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Old page wikitext, before the edit (old_wikitext) | 'In [[linear algebra]], a '''Hankel matrix''' (or '''[[catalecticant]] matrix'''), named after [[Hermann Hankel]], is a [[square matrix]] in which each ascending skew-diagonal from left to right is constant, e.g.:
<math display=block>\qquad\begin{bmatrix}
a & b & c & d & e \\
b & c & d & e & f \\
c & d & e & f & g \\
d & e & f & g & h \\
e & f & g & h & i \\
\end{bmatrix}.</math>
More generally, a '''Hankel matrix''' is any <math>n \times n</math> matrix <math>A</math> of the form
<math display=block>A = \begin{bmatrix}
a_{0} & a_{1} & a_{2} & \ldots & \ldots &a_{n-1} \\
a_{1} & a_2 & & & &\vdots \\
a_{2} & & & & & \vdots \\
\vdots & & & & & a_{2n-4}\\
\vdots & & & & a_{2n-4}& a_{2n-3} \\
a_{n-1} & \ldots & \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2}
\end{bmatrix}.</math>
In terms of the components, if the <math>i,j</math> element of <math>A</math> is denoted with <math>A_{ij}</math>, and assuming <math>i\le j</math>, then we have <math>A_{i,j} = A_{i+k,j-k}</math> for all <math>k = 0,...,j-i.</math>
==Properties==
* The Hankel matrix is a [[symmetric matrix]].
* Let <math>J_n</math> be the <math>n \times n</math> [[exchange matrix]]. If <math>H</math> is a <math>m \times n</math> Hankel matrix, then <math>H = T J_n</math> where <math>T</math> is a <math>m \times n</math> [[Toeplitz matrix]].
** If <math>T</math> is [[real number|real]] symmetric, then <math>H = T J_n</math> will have the same [[eigenvalue]]s as <math>T</math> up to sign.<ref name="simax1">{{cite journal | last = Yasuda | first = M. | title = A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 25 | issue = 3 | pages = 601–605 | year = 2003 | doi = 10.1137/S0895479802418835}}</ref>
* The [[Hilbert matrix]] is an example of a Hankel matrix.
==Hankel operator==
A Hankel [[operator (mathematics)|operator]] on a [[Hilbert space]] is one whose matrix is a (possibly infinite) Hankel matrix with respect to an [[orthonormal basis]]. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix <math>A </math> must satisfy, for all rows <math>i</math> and columns <math>j</math>, <math>(A_{i,j})_{i,j \ge 1}</math>. Note that every entry <math>A_{i,j}</math> depends only on <math>i+j</math>.
Let the corresponding '''Hankel Operator''' be <math>H_\alpha</math>. Given a Hankel matrix <math>A</math>, the corresponding Hankel operator is then defined as <math>H_\alpha(u)= Au</math>.
We are often interested in Hankel operators <math>H_\alpha: \ell^{2}\left(\mathbb{Z}^{+} \cup\{0\}\right) \rightarrow \ell^{2}\left(\mathbb{Z}^{+} \cup\{0\}\right)</math> over the Hilbert space <math>\ell^{2}(\mathbf Z) </math>, the space of square integrable bilateral [[complex number|complex]] [[sequence]]s. For any <math>u \in \ell^{2}(\mathbf Z)</math>, we have
<math display=block>\|u\|_{\ell^{2}(z)}^{2} = \sum_{n=-\infty}^{\infty}\left|u_{n}\right|^{2}</math>
We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests [[singular value decomposition]] as a possible technique to approximate the action of the operator.
Note that the matrix <math>A</math> does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.
The [[determinant]] of a Hankel matrix is called a [[catalecticant]].
==Hankel matrix transform==
{{Distinguish|Hankel transform}}
The '''Hankel matrix transform''', or simply '''Hankel transform''', produces the sequence of the [[determinant]]s of the Hankel matrices formed from the given sequence. Namely, the sequence <math>\{h_n\}_{n\ge 0}</math> is the Hankel transform of the sequence <math>\{b_n\}_{n\ge 0}</math> when
<math display=block>h_n = \det (b_{i+j-2})_{1 \le i,j \le n+1}.</math>
The Hankel transform is invariant under the [[binomial transform]] of a sequence. That is, if one writes
<math display=block>c_n = \sum_{k=0}^n {n \choose k} b_k</math>
as the binomial transform of the sequence <math>\{b_n\}</math>, then one has
<math display=block>\det (b_{i+j-2})_{1 \le i,j \le n+1} = \det (c_{i+j-2})_{1 \le i,j \le n+1}.</math>
== Applications of Hankel matrices ==
Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or [[hidden Markov model]] is desired.<ref>{{cite book |first=Masanao |last=Aoki |author-link=Masanao Aoki |chapter=Prediction of Time Series |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=38–47 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA38 }}</ref> The [[singular value decomposition]] of the Hankel matrix provides a means of computing the ''A'', ''B'', and ''C'' matrices which define the state-space realization.<ref>{{cite book |first=Masanao |last=Aoki |chapter=Rank determination of Hankel matrices |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=67–68 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA67 }}</ref> The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.
=== Method of moments for polynomial distributions ===
The [[Method of moments (statistics)|method of moments]] applied to polynomial distributions results in a Hankel matrix that needs to be [[inverse matrix|inverted]] in order to obtain the weight parameters of the polynomial distribution approximation.<ref name="PolyD2">J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". PLoS ONE 12(4): e0174573. https://doi.org/10.1371/journal.pone.0174573</ref>
=== Positive Hankel matrices and the Hamburger moment problems ===
{{Further|Hamburger moment problem}}
==See also==
* [[Toeplitz matrix]], an "upside down" (i.e., row-reversed) Hankel matrix
* [[Cauchy matrix]]
* [[Vandermonde matrix]]
== Notes ==
{{Reflist}}
== References ==
*[[Richard P. Brent|Brent R.P.]] (1999), "Stability of fast algorithms for structured linear systems", ''Fast Reliable Algorithms for Matrices with Structure'' (editors—T. Kailath, A.H. Sayed), ch.4 ([[Society for Industrial and Applied Mathematics|SIAM]]).
* {{cite book | title=Structured matrices and polynomials: unified superfast algorithms | author=Victor Y. Pan | author-link=Victor Pan | publisher=[[Birkhäuser]] | year=2001 | isbn=0817642404 }}
* {{cite book | title=An introduction to Hankel operators | author=J.R. Partington | author-link=Jonathan Partington | series=LMS Student Texts | volume=13 | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-36791-3 }}
{{Matrix classes}}
{{Authority control}}
[[Category:Matrices]]
[[Category:Transforms]]' |
New page wikitext, after the edit (new_wikitext) | 'In [[linear algebra]], a '''Hankel matrix''' (or '''[[catalecticant]] matrix'''), named after [[Hermann Hankel]], is a [[square matrix]] in which each ascending skew-diagonal from left to right is constant, e.g.:
<math display=block>\qquad\begin{bmatrix}
a & b & c & d & e \\
b & c & d & e & f \\
c & d & e & f & g \\
d & e & f & g & h \\
e & f & g & h & i \\
\end{bmatrix}.</math>
More generally, a '''Hankel matrix''' is any <math>n \times n</math> matrix <math>A</math> of the form
<math display=block>A = \begin{bmatrix}
a_{0} & a_{1} & a_{2} & \ldots & \ldots &a_{n-1} \\
a_{1} & a_2 & & & &\vdots \\
a_{2} & & & & & \vdots \\
\vdots & & & & & a_{2n-4}\\
\vdots & & & & a_{2n-4}& a_{2n-3} \\
a_{n-1} & \ldots & \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2}
\end{bmatrix}.</math>
In terms of the components, if the <math>i,j</math> element of <math>A</math> is denoted with <math>A_{ij}</math>, and assuming <math>i\le j</math>, then we have <math>A_{i,j} = A_{i+k,j-k}</math> for all <math>k = 0,...,j-i.</math>
==Properties==
* Any Hankel matrix is [[symmetric matrix|symmetric]].
* Let <math>J_n</math> be the <math>n \times n</math> [[exchange matrix]]. If <math>H</math> is a <math>m \times n</math> Hankel matrix, then <math>H = T J_n</math> where <math>T</math> is a <math>m \times n</math> [[Toeplitz matrix]].
** If <math>T</math> is [[real number|real]] symmetric, then <math>H = T J_n</math> will have the same [[eigenvalue]]s as <math>T</math> up to sign.<ref name="simax1">{{cite journal | last = Yasuda | first = M. | title = A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 25 | issue = 3 | pages = 601–605 | year = 2003 | doi = 10.1137/S0895479802418835}}</ref>
* The [[Hilbert matrix]] is an example of a Hankel matrix.
==Relation to formal Laurent series==
Hankel matrices are closely related to [[formal Laurent series]].<ref>{{harvnb|Fuhrmann|2012|§8.3}}</ref> In fact, such a series <math>f(z) = \sum_{n=-\infty}^N a_n z^n</math> gives rise to a linear map, referred to as a ''Hankel operator''
:<math>H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf C[[z^{-1}]],</math>
which takes a [[polynomial]] <math>g \in \mathbf C[z]</math> and sends it to the product <math>f g</math>, but discards all powers of <math>z</math> with a non-negative exponent, so as to give an element in <math>z^{-1} \mathbf C[[z^{-1}]]</math>, the [[formal power series]] with strictly negative exponents. The map <math>H_f</math> is in a natural way <math>\mathbf C[z]</math>-linear, and its matrix with respect to the elements <math>1, z, z^2, \dots \in \mathbf C[z]</math> and <math>z^{-1}, z^{-2}, \dots \in z^{-1}\mathbf C[[z^{-1}]]</math> is the Hankel matrix
:<math display=block>\begin{bmatrix}
a_1 & a_2 & \ldots \\
a_2 & a_3 & \ldots \\
a_3 & a_4 & \ldots \\
\vdots
\end{bmatrix}.</math>
Any Hankel matrix arises in such a way. A theorem due to [[Kronecker]] says that the [[Rank (linear algebra)|rank]] of this matrix is finite precisely if <math>f</math> is a [[rational function]], i.e., a fraction of two polynomials <math>f = \frac {p(z)}{q(z)}</math>.
==Hankel operator==
A Hankel [[operator (mathematics)|operator]] on a [[Hilbert space]] is one whose matrix is a (possibly infinite) Hankel matrix with respect to an [[orthonormal basis]]. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix <math>A </math> must satisfy, for all rows <math>i</math> and columns <math>j</math>, <math>(A_{i,j})_{i,j \ge 1}</math>. Note that every entry <math>A_{i,j}</math> depends only on <math>i+j</math>.
Let the corresponding '''Hankel Operator''' be <math>H_\alpha</math>. Given a Hankel matrix <math>A</math>, the corresponding Hankel operator is then defined as <math>H_\alpha(u)= Au</math>.
We are often interested in Hankel operators <math>H_\alpha: \ell^{2}\left(\mathbb{Z}^{+} \cup\{0\}\right) \rightarrow \ell^{2}\left(\mathbb{Z}^{+} \cup\{0\}\right)</math> over the Hilbert space <math>\ell^{2}(\mathbf Z) </math>, the space of square integrable bilateral [[complex number|complex]] [[sequence]]s. For any <math>u \in \ell^{2}(\mathbf Z)</math>, we have
<math display=block>\|u\|_{\ell^{2}(z)}^{2} = \sum_{n=-\infty}^{\infty}\left|u_{n}\right|^{2}</math>
We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests [[singular value decomposition]] as a possible technique to approximate the action of the operator.
Note that the matrix <math>A</math> does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.
The [[determinant]] of a Hankel matrix is called a [[catalecticant]].
==Hankel matrix transform==
{{Distinguish|Hankel transform}}
The '''Hankel matrix transform''', or simply '''Hankel transform''', produces the sequence of the [[determinant]]s of the Hankel matrices formed from the given sequence. Namely, the sequence <math>\{h_n\}_{n\ge 0}</math> is the Hankel transform of the sequence <math>\{b_n\}_{n\ge 0}</math> when
<math display=block>h_n = \det (b_{i+j-2})_{1 \le i,j \le n+1}.</math>
The Hankel transform is invariant under the [[binomial transform]] of a sequence. That is, if one writes
<math display=block>c_n = \sum_{k=0}^n {n \choose k} b_k</math>
as the binomial transform of the sequence <math>\{b_n\}</math>, then one has
<math display=block>\det (b_{i+j-2})_{1 \le i,j \le n+1} = \det (c_{i+j-2})_{1 \le i,j \le n+1}.</math>
== Applications of Hankel matrices ==
Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or [[hidden Markov model]] is desired.<ref>{{cite book |first=Masanao |last=Aoki |author-link=Masanao Aoki |chapter=Prediction of Time Series |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=38–47 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA38 }}</ref> The [[singular value decomposition]] of the Hankel matrix provides a means of computing the ''A'', ''B'', and ''C'' matrices which define the state-space realization.<ref>{{cite book |first=Masanao |last=Aoki |chapter=Rank determination of Hankel matrices |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=67–68 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA67 }}</ref> The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.
=== Method of moments for polynomial distributions ===
The [[Method of moments (statistics)|method of moments]] applied to polynomial distributions results in a Hankel matrix that needs to be [[inverse matrix|inverted]] in order to obtain the weight parameters of the polynomial distribution approximation.<ref name="PolyD2">J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". PLoS ONE 12(4): e0174573. https://doi.org/10.1371/journal.pone.0174573</ref>
=== Positive Hankel matrices and the Hamburger moment problems ===
{{Further|Hamburger moment problem}}
==See also==
* [[Toeplitz matrix]], an "upside down" (i.e., row-reversed) Hankel matrix
* [[Cauchy matrix]]
* [[Vandermonde matrix]]
== Notes ==
{{Reflist}}
== References ==
*[[Richard P. Brent|Brent R.P.]] (1999), "Stability of fast algorithms for structured linear systems", ''Fast Reliable Algorithms for Matrices with Structure'' (editors—T. Kailath, A.H. Sayed), ch.4 ([[Society for Industrial and Applied Mathematics|SIAM]]).
*{{cite book
| last = Fuhrmann
| first = Paul A.
| title = A polynomial approach to linear algebra
| edition = 2
| series = Universitext
| year = 2012
| publisher = Springer
| location = New York, NY
| isbn = 978-1-4614-0337-1
| doi = 10.1007/978-1-4614-0338-8
| zbl = 05934737
}}
* {{cite book | title=Structured matrices and polynomials: unified superfast algorithms | author=Victor Y. Pan | author-link=Victor Pan | publisher=[[Birkhäuser]] | year=2001 | isbn=0817642404 }}
* {{cite book | title=An introduction to Hankel operators | author=J.R. Partington | author-link=Jonathan Partington | series=LMS Student Texts | volume=13 | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-36791-3 }}
{{Matrix classes}}
{{Authority control}}
[[Category:Matrices]]
[[Category:Transforms]]' |
Lines added in edit (added_lines) | [
0 => '* Any Hankel matrix is [[symmetric matrix|symmetric]].',
1 => '',
2 => '==Relation to formal Laurent series==',
3 => 'Hankel matrices are closely related to [[formal Laurent series]].<ref>{{harvnb|Fuhrmann|2012|§8.3}}</ref> In fact, such a series <math>f(z) = \sum_{n=-\infty}^N a_n z^n</math> gives rise to a linear map, referred to as a ''Hankel operator''',
4 => ':<math>H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf C[[z^{-1}]],</math>',
5 => 'which takes a [[polynomial]] <math>g \in \mathbf C[z]</math> and sends it to the product <math>f g</math>, but discards all powers of <math>z</math> with a non-negative exponent, so as to give an element in <math>z^{-1} \mathbf C[[z^{-1}]]</math>, the [[formal power series]] with strictly negative exponents. The map <math>H_f</math> is in a natural way <math>\mathbf C[z]</math>-linear, and its matrix with respect to the elements <math>1, z, z^2, \dots \in \mathbf C[z]</math> and <math>z^{-1}, z^{-2}, \dots \in z^{-1}\mathbf C[[z^{-1}]]</math> is the Hankel matrix ',
6 => ':<math display=block>\begin{bmatrix}',
7 => ' a_1 & a_2 & \ldots \\',
8 => ' a_2 & a_3 & \ldots \\',
9 => ' a_3 & a_4 & \ldots \\ ',
10 => ' \vdots',
11 => '\end{bmatrix}.</math>',
12 => 'Any Hankel matrix arises in such a way. A theorem due to [[Kronecker]] says that the [[Rank (linear algebra)|rank]] of this matrix is finite precisely if <math>f</math> is a [[rational function]], i.e., a fraction of two polynomials <math>f = \frac {p(z)}{q(z)}</math>.',
13 => '*{{cite book',
14 => ' | last = Fuhrmann',
15 => ' | first = Paul A.',
16 => ' | title = A polynomial approach to linear algebra',
17 => ' | edition = 2',
18 => ' | series = Universitext',
19 => ' | year = 2012',
20 => ' | publisher = Springer',
21 => ' | location = New York, NY',
22 => ' | isbn = 978-1-4614-0337-1',
23 => ' | doi = 10.1007/978-1-4614-0338-8',
24 => ' | zbl = 05934737',
25 => '}}',
26 => ''
] |
Parsed HTML source of the new revision (new_html) | '<div class="mw-parser-output"><p>In <a href="/enwiki/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, a <b>Hankel matrix</b> (or <b><a href="/enwiki/wiki/Catalecticant" title="Catalecticant">catalecticant</a> matrix</b>), named after <a href="/enwiki/wiki/Hermann_Hankel" title="Hermann Hankel">Hermann Hankel</a>, is a <a href="/enwiki/wiki/Square_matrix" title="Square matrix">square matrix</a> in which each ascending skew-diagonal from left to right is constant, e.g.:
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.}">
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</p><p>More generally, a <b>Hankel matrix</b> is any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="n\times n"></span> matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> of the form
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &\ldots &a_{n-1}\\a_{1}&a_{2}&&&&\vdots \\a_{2}&&&&&\vdots \\\vdots &&&&&a_{2n-4}\\\vdots &&&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd />
<mtd />
<mtd />
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mtd>
<mtd />
<mtd />
<mtd />
<mtd />
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd />
<mtd />
<mtd />
<mtd />
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>⋮<!-- ⋮ --></mo>
</mtd>
<mtd />
<mtd />
<mtd />
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<mo>…<!-- … --></mo>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &\ldots &a_{n-1}\\a_{1}&a_{2}&&&&\vdots \\a_{2}&&&&&\vdots \\\vdots &&&&&a_{2n-4}\\\vdots &&&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3e3343a5982e2f22f933d6e291ecbe7b36977525" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -11.338ex; width:47.061ex; height:23.843ex;" alt="{\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &\ldots &a_{n-1}\\a_{1}&a_{2}&&&&\vdots \\a_{2}&&&&&\vdots \\\vdots &&&&&a_{2n-4}\\\vdots &&&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}"></div>
</p><p>In terms of the components, if the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i,j}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f4cbf8bbc622154cda8208d6e339495fe16a1f9a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="i,j"></span> element of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> is denoted with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{ij}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8272b28f5aae6dbb8d6f829d58bab353b21bde20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:3.22ex; height:2.843ex;" alt="A_{ij}"></span>, and assuming <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\leq j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>≤<!-- ≤ --></mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i\leq j}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/894ab6e9c9afcfea7d9370399cebe1557bdf9b2e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="i\leq j"></span>, then we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i,j}=A_{i+k,j-k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>+</mo>
<mi>k</mi>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{i,j}=A_{i+k,j-k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a0f0e4c1091cb8d6331457ec9f76f8e6b874f84f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:14.724ex; height:2.843ex;" alt="{\displaystyle A_{i,j}=A_{i+k,j-k}}"></span> for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,...,j-i.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mi>i</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k=0,...,j-i.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3617d0e300c55f527282e9b941cdc96d4e926a00" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:15.89ex; height:2.509ex;" alt="{\displaystyle k=0,...,j-i.}"></span>
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Properties"><span class="tocnumber">1</span> <span class="toctext">Properties</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Relation_to_formal_Laurent_series"><span class="tocnumber">2</span> <span class="toctext">Relation to formal Laurent series</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Hankel_operator"><span class="tocnumber">3</span> <span class="toctext">Hankel operator</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Hankel_matrix_transform"><span class="tocnumber">4</span> <span class="toctext">Hankel matrix transform</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Applications_of_Hankel_matrices"><span class="tocnumber">5</span> <span class="toctext">Applications of Hankel matrices</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Method_of_moments_for_polynomial_distributions"><span class="tocnumber">5.1</span> <span class="toctext">Method of moments for polynomial distributions</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Positive_Hankel_matrices_and_the_Hamburger_moment_problems"><span class="tocnumber">5.2</span> <span class="toctext">Positive Hankel matrices and the Hamburger moment problems</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-8"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-9"><a href="#Notes"><span class="tocnumber">7</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#References"><span class="tocnumber">8</span> <span class="toctext">References</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=1" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li>Any Hankel matrix is <a href="/enwiki/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>.</li>
<li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/22b79b47e08e4e16510d309639ef56a24c28696c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.509ex;" alt="J_n"></span> be the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="n\times n"></span> <a href="/enwiki/wiki/Exchange_matrix" title="Exchange matrix">exchange matrix</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="H"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="m\times n"></span> Hankel matrix, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=TJ_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>=</mo>
<mi>T</mi>
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H=TJ_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d76572040aaa8196f0279a8029d75a2edb903696" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.307ex; height:2.509ex;" alt="{\displaystyle H=TJ_{n}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="m\times n"></span> <a href="/enwiki/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a>.
<ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"></span> is <a href="/enwiki/wiki/Real_number" title="Real number">real</a> symmetric, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=TJ_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>H</mi>
<mo>=</mo>
<mi>T</mi>
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H=TJ_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/d76572040aaa8196f0279a8029d75a2edb903696" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.307ex; height:2.509ex;" alt="{\displaystyle H=TJ_{n}}"></span> will have the same <a href="/enwiki/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="T"></span> up to sign.<sup id="cite_ref-simax1_1-0" class="reference"><a href="#cite_note-simax1-1">[1]</a></sup></li></ul></li>
<li>The <a href="/enwiki/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert matrix</a> is an example of a Hankel matrix.</li></ul>
<h2><span class="mw-headline" id="Relation_to_formal_Laurent_series">Relation to formal Laurent series</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=2" title="Edit section: Relation to formal Laurent series">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Hankel matrices are closely related to <a href="/enwiki/wiki/Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup> In fact, such a series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>N</mi>
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</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/99f01b2bfc27bd94a8fb06ae5ef8158131f74bec" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:17.605ex; height:7.343ex;" alt="{\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n}}"></span> gives rise to a linear map, referred to as a <i>Hankel operator</i>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2e44553b7947438bc382d65352fb94867a82e841" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:25.041ex; height:3.343ex;" alt="{\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]],}"></span></dd></dl>
<p>which takes a <a href="/enwiki/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in \mathbf {C} [z]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g\in \mathbf {C} [z]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3df7d024d0b972a46da42c84aeff389ff5458f1f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.27ex; height:2.843ex;" alt="{\displaystyle g\in \mathbf {C} [z]}"></span> and sends it to the product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle fg}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}"></span>, but discards all powers of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="z"></span> with a non-negative exponent, so as to give an element in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/5a964b9b781a86975a982d8911e5bef310129193" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.365ex; height:3.176ex;" alt="{\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}"></span>, the <a href="/enwiki/wiki/Formal_power_series" title="Formal power series">formal power series</a> with strictly negative exponents. The map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{f}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{f}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8d9e0161729e04d187152ed5e25cca64312eed39" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:3.068ex; height:2.843ex;" alt="H_{f}"></span> is in a natural way <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} [z]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {C} [z]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4df688b72293d8ac85f15267559929137e35ff1c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.313ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} [z]}"></span>-linear, and its matrix with respect to the elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mi>z</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2814851a432ce68c94aab843fac3ba4882bbd3a6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:17.374ex; height:3.176ex;" alt="{\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>,</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msup>
<mo>,</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<mo stretchy="false">[</mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">]</mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bbf789a2c0db73be54af3b1919900b73cf98bad9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:25.843ex; height:3.176ex;" alt="{\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}"></span> is the Hankel matrix
</p>
<dl><dd><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots \end{bmatrix}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
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<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mtd>
<msub>
<mi>a</mi>
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<mn>2</mn>
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</msub>
</mtd>
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<mtr>
<mtd>
<msub>
<mi>a</mi>
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<mn>3</mn>
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<mtd>
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<mi>a</mi>
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<mn>3</mn>
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<mtd>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
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<mtd>
<mo>…<!-- … --></mo>
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<mtd>
<mo>⋮<!-- ⋮ --></mo>
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</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots \end{bmatrix}}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e9338c54f44d1c32b7b4eb4f062ff25cb45e4653" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -6.338ex; width:16.435ex; height:13.843ex;" alt="{\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots \end{bmatrix}}.}"></div></dd></dl>
<p>Any Hankel matrix arises in such a way. A theorem due to <a href="/enwiki/wiki/Kronecker" title="Kronecker">Kronecker</a> says that the <a href="/enwiki/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> of this matrix is finite precisely if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is a <a href="/enwiki/wiki/Rational_function" title="Rational function">rational function</a>, i.e., a fraction of two polynomials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {p(z)}{q(z)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>q</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f={\frac {p(z)}{q(z)}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/64c87f1dcc147288cee4a7c970ca69c3bc0631c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:9.28ex; height:6.509ex;" alt="{\displaystyle f={\frac {p(z)}{q(z)}}}"></span>.
</p>
<h2><span class="mw-headline" id="Hankel_operator">Hankel operator</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=3" title="Edit section: Hankel operator">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>A Hankel <a href="/enwiki/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> on a <a href="/enwiki/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> is one whose matrix is a (possibly infinite) Hankel matrix with respect to an <a href="/enwiki/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> must satisfy, for all rows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i"></span> and columns <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="j"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A_{i,j})_{i,j\geq 1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>≥<!-- ≥ --></mo>
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (A_{i,j})_{i,j\geq 1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bee9a690a4c400a164232bfe02d74114894c4e1b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.522ex; height:3.009ex;" alt="(A_{i,j})_{i,j\geq 1}"></span>. Note that every entry <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i,j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A_{i,j}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/4f379f42961dd620c7a05dc1c538117ec105877d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:3.678ex; height:2.843ex;" alt="A_{i,j}"></span> depends only on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i+j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i+j}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ea2c2b111f44787702a0e16807f7d66b541d1c59" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.601ex; height:2.509ex;" alt="i+j"></span>.
</p><p>Let the corresponding <b>Hankel Operator</b> be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{\alpha }}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c4ca84c2be95eec6d5c19bfc17c5bcd2ecf16724" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.215ex; height:2.509ex;" alt="H_{\alpha }"></span>. Given a Hankel matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span>, the corresponding Hankel operator is then defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\alpha }(u)=Au}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>u</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{\alpha }(u)=Au}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c123423f2ba9376c09bd34f400ecf840cca91ad7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.526ex; height:2.843ex;" alt="{\displaystyle H_{\alpha }(u)=Au}"></span>.
</p><p>We are often interested in Hankel operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\alpha }:\ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)\rightarrow \ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<mo>:</mo>
<msup>
<mi>ℓ<!-- ℓ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo>∪<!-- ∪ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mi>ℓ<!-- ℓ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
<mo>∪<!-- ∪ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H_{\alpha }:\ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)\rightarrow \ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b08bec9d11ae2d4d08215704d6f293eb6c31cfcf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:36.11ex; height:3.343ex;" alt="{\displaystyle H_{\alpha }:\ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)\rightarrow \ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)}"></span> over the Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ^{2}(\mathbf {Z} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>ℓ<!-- ℓ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Z</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ell ^{2}(\mathbf {Z} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e9b1009697fbe680f2d1f8026d8bcc734abd3fbd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:5.467ex; height:3.176ex;" alt="{\displaystyle \ell ^{2}(\mathbf {Z} )}"></span>, the space of square integrable bilateral <a href="/enwiki/wiki/Complex_number" title="Complex number">complex</a> <a href="/enwiki/wiki/Sequence" title="Sequence">sequences</a>. For any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \ell ^{2}(\mathbf {Z} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>ℓ<!-- ℓ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">Z</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u\in \ell ^{2}(\mathbf {Z} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8606d561b473a9d3274663af342b8fd7a893d487" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.637ex; height:3.176ex;" alt="{\displaystyle u\in \ell ^{2}(\mathbf {Z} )}"></span>, we have
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|_{\ell ^{2}(z)}^{2}=\sum _{n=-\infty }^{\infty }\left|u_{n}\right|^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mi>u</mi>
<msubsup>
<mo fence="false" stretchy="false">‖<!-- ‖ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>ℓ<!-- ℓ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
</munderover>
<msup>
<mrow>
<mo>|</mo>
<msub>
<mi>u</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \|u\|_{\ell ^{2}(z)}^{2}=\sum _{n=-\infty }^{\infty }\left|u_{n}\right|^{2}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/93536a0ce62c85c6ab0b2feeebfcd437de0eaaee" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.005ex; width:21.021ex; height:6.843ex;" alt="{\displaystyle \|u\|_{\ell ^{2}(z)}^{2}=\sum _{n=-\infty }^{\infty }\left|u_{n}\right|^{2}}"></div>
</p><p>We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests <a href="/enwiki/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> as a possible technique to approximate the action of the operator.
</p><p>Note that the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.
</p><p>The <a href="/enwiki/wiki/Determinant" title="Determinant">determinant</a> of a Hankel matrix is called a <a href="/enwiki/wiki/Catalecticant" title="Catalecticant">catalecticant</a>.
</p>
<h2><span class="mw-headline" id="Hankel_matrix_transform">Hankel matrix transform</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=4" title="Edit section: Hankel matrix transform">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/enwiki/wiki/Hankel_transform" title="Hankel transform">Hankel transform</a>.</div>
<p>The <b>Hankel matrix transform</b>, or simply <b>Hankel transform</b>, produces the sequence of the <a href="/enwiki/wiki/Determinant" title="Determinant">determinants</a> of the Hankel matrices formed from the given sequence. Namely, the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{h_{n}\}_{n\geq 0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{h_{n}\}_{n\geq 0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9f4ec477137d15a57dd2bafcdfdaf69fc19107bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.201ex; height:2.843ex;" alt="\{h_{n}\}_{n\geq 0}"></span> is the Hankel transform of the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}_{n\geq 0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<msub>
<mo fence="false" stretchy="false">}</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}_{n\geq 0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/3d3c33aa493f39c31107ac4a080f490bc9b4ea3c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.86ex; height:2.843ex;" alt="\{b_{n}\}_{n\geq 0}"></span> when
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>h</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mo movablelimits="true" form="prefix">det</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>≤<!-- ≤ --></mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>≤<!-- ≤ --></mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/cf0c910033fe6c6e8c917ef2681eda7bdac43464" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -1.005ex; width:25.596ex; height:3.009ex;" alt="{\displaystyle h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.}"></div>
</p><p>The Hankel transform is invariant under the <a href="/enwiki/wiki/Binomial_transform" title="Binomial transform">binomial transform</a> of a sequence. That is, if one writes
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mi>k</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6216b5423bc58e9f10c200d52a953ea8f3cbc4d0" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -3.171ex; width:15.968ex; height:7.009ex;" alt="{\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}"></div>
</p><p>as the binomial transform of the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{n}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{b_{n}\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/cad2485b9672375982ec521a53ee5a4104001a15" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="\{b_{n}\}"></span>, then one has
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ed83a3e22809a5cb5485fc63b8f896b1500f3e46" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -1.005ex; width:42.34ex; height:3.009ex;" alt="{\displaystyle \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.}"></div>
</p>
<h2><span class="mw-headline" id="Applications_of_Hankel_matrices">Applications of Hankel matrices</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=5" title="Edit section: Applications of Hankel matrices">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or <a href="/enwiki/wiki/Hidden_Markov_model" title="Hidden Markov model">hidden Markov model</a> is desired.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> The <a href="/enwiki/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> of the Hankel matrix provides a means of computing the <i>A</i>, <i>B</i>, and <i>C</i> matrices which define the state-space realization.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.
</p>
<h3><span class="mw-headline" id="Method_of_moments_for_polynomial_distributions">Method of moments for polynomial distributions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=6" title="Edit section: Method of moments for polynomial distributions">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The <a href="/enwiki/wiki/Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a> applied to polynomial distributions results in a Hankel matrix that needs to be <a href="/enwiki/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverted</a> in order to obtain the weight parameters of the polynomial distribution approximation.<sup id="cite_ref-PolyD2_5-0" class="reference"><a href="#cite_note-PolyD2-5">[5]</a></sup>
</p>
<h3><span class="mw-headline" id="Positive_Hankel_matrices_and_the_Hamburger_moment_problems">Positive Hankel matrices and the Hamburger moment problems</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=7" title="Edit section: Positive Hankel matrices and the Hamburger moment problems">edit</a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/enwiki/wiki/Hamburger_moment_problem" title="Hamburger moment problem">Hamburger moment problem</a></div>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=8" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a href="/enwiki/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a>, an "upside down" (i.e., row-reversed) Hankel matrix</li>
<li><a href="/enwiki/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy matrix</a></li>
<li><a href="/enwiki/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a></li></ul>
<h2><span class="mw-headline" id="Notes">Notes</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=9" title="Edit section: Notes">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-simax1-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-simax1_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1133582631">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("/upwiki/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("/upwiki/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("/upwiki/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("/upwiki/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFYasuda2003" class="citation journal cs1">Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". <i>SIAM J. Matrix Anal. Appl</i>. <b>25</b> (3): 601–605. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0895479802418835">10.1137/S0895479802418835</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+J.+Matrix+Anal.+Appl.&rft.atitle=A+Spectral+Characterization+of+Hermitian+Centrosymmetric+and+Hermitian+Skew-Centrosymmetric+K-Matrices&rft.volume=25&rft.issue=3&rft.pages=601-605&rft.date=2003&rft_id=info%3Adoi%2F10.1137%2FS0895479802418835&rft.aulast=Yasuda&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFFuhrmann2012§8.3">Fuhrmann, 2012 & §8.3</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFFuhrmann2012§8.3 (<a href="/enwiki/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAoki1983" class="citation book cs1"><a href="/enwiki/wiki/Masanao_Aoki" title="Masanao Aoki">Aoki, Masanao</a> (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA38">"Prediction of Time Series"</a>. <i>Notes on Economic Time Series Analysis : System Theoretic Perspectives</i>. New York: Springer. pp. 38–47. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-387-12696-1" title="Special:BookSources/0-387-12696-1"><bdi>0-387-12696-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Prediction+of+Time+Series&rft.btitle=Notes+on+Economic+Time+Series+Analysis+%3A+System+Theoretic+Perspectives&rft.place=New+York&rft.pages=38-47&rft.pub=Springer&rft.date=1983&rft.isbn=0-387-12696-1&rft.aulast=Aoki&rft.aufirst=Masanao&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dl_LsCAAAQBAJ%26pg%3DPA38&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAoki1983" class="citation book cs1">Aoki, Masanao (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA67">"Rank determination of Hankel matrices"</a>. <i>Notes on Economic Time Series Analysis : System Theoretic Perspectives</i>. New York: Springer. pp. 67–68. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-387-12696-1" title="Special:BookSources/0-387-12696-1"><bdi>0-387-12696-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rank+determination+of+Hankel+matrices&rft.btitle=Notes+on+Economic+Time+Series+Analysis+%3A+System+Theoretic+Perspectives&rft.place=New+York&rft.pages=67-68&rft.pub=Springer&rft.date=1983&rft.isbn=0-387-12696-1&rft.aulast=Aoki&rft.aufirst=Masanao&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dl_LsCAAAQBAJ%26pg%3DPA67&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span></span>
</li>
<li id="cite_note-PolyD2-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-PolyD2_5-0">^</a></b></span> <span class="reference-text">J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". PLoS ONE 12(4): e0174573. <a rel="nofollow" class="external free" href="https://doi.org/10.1371/journal.pone.0174573">https://doi.org/10.1371/journal.pone.0174573</a></span>
</li>
</ol></div></div>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/enwiki/w/index.php?title=Hankel_matrix&action=edit&section=10" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a href="/enwiki/wiki/Richard_P._Brent" title="Richard P. Brent">Brent R.P.</a> (1999), "Stability of fast algorithms for structured linear systems", <i>Fast Reliable Algorithms for Matrices with Structure</i> (editors—T. Kailath, A.H. Sayed), ch.4 (<a href="/enwiki/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>).</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFFuhrmann2012" class="citation book cs1">Fuhrmann, Paul A. (2012). <i>A polynomial approach to linear algebra</i>. Universitext (2 ed.). New York, NY: Springer. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-0338-8">10.1007/978-1-4614-0338-8</a>. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-1-4614-0337-1" title="Special:BookSources/978-1-4614-0337-1"><bdi>978-1-4614-0337-1</bdi></a>. <a href="/enwiki/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:05934737">05934737</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+polynomial+approach+to+linear+algebra&rft.place=New+York%2C+NY&rft.series=Universitext&rft.edition=2&rft.pub=Springer&rft.date=2012&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A05934737%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-0338-8&rft.isbn=978-1-4614-0337-1&rft.aulast=Fuhrmann&rft.aufirst=Paul+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/enwiki/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: Zbl (<a href="/enwiki/wiki/Category:CS1_maint:_Zbl" title="Category:CS1 maint: Zbl">link</a>)</span></li></ul>
<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFVictor_Y._Pan2001" class="citation book cs1"><a href="/enwiki/wiki/Victor_Pan" title="Victor Pan">Victor Y. Pan</a> (2001). <i>Structured matrices and polynomials: unified superfast algorithms</i>. <a href="/enwiki/wiki/Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0817642404" title="Special:BookSources/0817642404"><bdi>0817642404</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structured+matrices+and+polynomials%3A+unified+superfast+algorithms&rft.pub=Birkh%C3%A4user&rft.date=2001&rft.isbn=0817642404&rft.au=Victor+Y.+Pan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJ.R._Partington1988" class="citation book cs1"><a href="/enwiki/wiki/Jonathan_Partington" title="Jonathan Partington">J.R. Partington</a> (1988). <i>An introduction to Hankel operators</i>. LMS Student Texts. Vol. 13. <a href="/enwiki/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-521-36791-3" title="Special:BookSources/0-521-36791-3"><bdi>0-521-36791-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+Hankel+operators&rft.series=LMS+Student+Texts&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=0-521-36791-3&rft.au=J.R.+Partington&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHankel+matrix" class="Z3988"></span></li></ul>
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transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-talk"><a href="/enwiki/wiki/Template_talk:Matrix_classes" title="Template talk:Matrix classes"><abbr title="Discuss this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-edit"><a class="external text" href="https://en.wikipedia.org/enwiki/w/index.php?title=Template:Matrix_classes&action=edit"><abbr title="Edit this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="Matrix_classes" style="font-size:114%;margin:0 4em"><a href="/enwiki/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicitly constrained entries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Alternant_matrix" title="Alternant matrix">Alternant</a></li>
<li><a href="/enwiki/wiki/Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal</a></li>
<li><a href="/enwiki/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li>
<li><a href="/enwiki/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li>
<li><a href="/enwiki/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li>
<li><a href="/enwiki/wiki/Band_matrix" title="Band matrix">Band</a></li>
<li><a href="/enwiki/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li>
<li><a href="/enwiki/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li>
<li><a href="/enwiki/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li>
<li><a href="/enwiki/wiki/Block_matrix" title="Block matrix">Block</a></li>
<li><a href="/enwiki/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li>
<li><a href="/enwiki/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li>
<li><a href="/enwiki/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li>
<li><a href="/enwiki/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li>
<li><a href="/enwiki/wiki/Conference_matrix" title="Conference matrix">Conference</a></li>
<li><a href="/enwiki/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li>
<li><a href="/enwiki/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li>
<li><a href="/enwiki/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li>
<li><a href="/enwiki/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li>
<li><a href="/enwiki/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li>
<li><a href="/enwiki/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li>
<li><a href="/enwiki/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li>
<li><a href="/enwiki/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li>
<li><a href="/enwiki/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li>
<li><a href="/enwiki/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li>
<li><a class="mw-selflink selflink">Hankel</a></li>
<li><a href="/enwiki/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li>
<li><a href="/enwiki/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li>
<li><a href="/enwiki/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li>
<li><a href="/enwiki/wiki/Integer_matrix" title="Integer matrix">Integer</a></li>
<li><a href="/enwiki/wiki/Logical_matrix" title="Logical matrix">Logical</a></li>
<li><a href="/enwiki/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li>
<li><a href="/enwiki/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li>
<li><a href="/enwiki/wiki/Moore_matrix" title="Moore matrix">Moore</a></li>
<li><a href="/enwiki/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li>
<li><a href="/enwiki/wiki/Pentadiagonal_matrix" title="Pentadiagonal matrix">Pentadiagonal</a></li>
<li><a href="/enwiki/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li>
<li><a href="/enwiki/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li>
<li><a href="/enwiki/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li>
<li><a href="/enwiki/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li>
<li><a href="/enwiki/wiki/Signature_matrix" title="Signature matrix">Signature</a></li>
<li><a href="/enwiki/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li>
<li><a href="/enwiki/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li>
<li><a href="/enwiki/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li>
<li><a href="/enwiki/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li>
<li><a href="/enwiki/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li>
<li><a href="/enwiki/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li>
<li><a href="/enwiki/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li>
<li><a href="/enwiki/wiki/Triangular_matrix" title="Triangular matrix">Triangular</a></li>
<li><a href="/enwiki/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li>
<li><a href="/enwiki/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li>
<li><a href="/enwiki/wiki/Walsh_matrix" title="Walsh matrix">Walsh</a></li>
<li><a href="/enwiki/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Exchange_matrix" title="Exchange matrix">Exchange</a></li>
<li><a href="/enwiki/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li>
<li><a href="/enwiki/wiki/Identity_matrix" title="Identity matrix">Identity</a></li>
<li><a href="/enwiki/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li>
<li><a href="/enwiki/wiki/Matrix_of_ones" title="Matrix of ones">Of ones</a></li>
<li><a href="/enwiki/wiki/Pascal_matrix" title="Pascal matrix">Pascal</a></li>
<li><a href="/enwiki/wiki/Pauli_matrices" title="Pauli matrices">Pauli</a></li>
<li><a href="/enwiki/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li>
<li><a href="/enwiki/wiki/Shift_matrix" title="Shift matrix">Shift</a></li>
<li><a href="/enwiki/wiki/Zero_matrix" title="Zero matrix">Zero</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="/enwiki/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Companion_matrix" title="Companion matrix">Companion</a></li>
<li><a href="/enwiki/wiki/Convergent_matrix" title="Convergent matrix">Convergent</a></li>
<li><a href="/enwiki/wiki/Defective_matrix" title="Defective matrix">Defective</a></li>
<li><a href="/enwiki/wiki/Definite_matrix" title="Definite matrix">Definite</a></li>
<li><a href="/enwiki/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li>
<li><a href="/enwiki/wiki/Hurwitz_matrix" title="Hurwitz matrix">Hurwitz</a></li>
<li><a href="/enwiki/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li>
<li><a href="/enwiki/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="/enwiki/wiki/Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="/enwiki/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Matrix_congruence" title="Matrix congruence">Congruent</a></li>
<li><a href="/enwiki/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="/enwiki/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li>
<li><a href="/enwiki/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li>
<li><a href="/enwiki/wiki/Involutory_matrix" title="Involutory matrix">Involutory</a></li>
<li><a href="/enwiki/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li>
<li><a href="/enwiki/wiki/Normal_matrix" title="Normal matrix">Normal</a></li>
<li><a href="/enwiki/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li>
<li><a href="/enwiki/wiki/Unimodular_matrix" title="Unimodular matrix">Unimodular</a></li>
<li><a href="/enwiki/wiki/Unipotent" title="Unipotent">Unipotent</a></li>
<li><a href="/enwiki/wiki/Unitary_matrix" title="Unitary matrix">Unitary</a></li>
<li><a href="/enwiki/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li>
<li><a href="/enwiki/wiki/Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li>
<li><a href="/enwiki/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li>
<li><a href="/enwiki/wiki/Augmented_matrix" title="Augmented matrix">Augmented</a></li>
<li><a href="/enwiki/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li>
<li><a href="/enwiki/wiki/Carleman_matrix" title="Carleman matrix">Carleman</a></li>
<li><a href="/enwiki/wiki/Cartan_matrix" title="Cartan matrix">Cartan</a></li>
<li><a href="/enwiki/wiki/Circulant_matrix" title="Circulant matrix">Circulant</a></li>
<li><a href="/enwiki/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li>
<li><a href="/enwiki/wiki/Commutation_matrix" title="Commutation matrix">Commutation</a></li>
<li><a href="/enwiki/wiki/Confusion_matrix" title="Confusion matrix">Confusion</a></li>
<li><a href="/enwiki/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li>
<li><a href="/enwiki/wiki/Distance_matrix" title="Distance matrix">Distance</a></li>
<li><a href="/enwiki/wiki/Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li>
<li><a href="/enwiki/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li>
<li><a href="/enwiki/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li>
<li><a href="/enwiki/wiki/Generator_matrix" title="Generator matrix">Generator</a></li>
<li><a href="/enwiki/wiki/Gram_matrix" title="Gram matrix">Gram</a></li>
<li><a href="/enwiki/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li>
<li><a href="/enwiki/wiki/Householder_transformation" title="Householder transformation">Householder</a></li>
<li><a href="/enwiki/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>
<li><a href="/enwiki/wiki/Moment_matrix" title="Moment matrix">Moment</a></li>
<li><a href="/enwiki/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li>
<li><a href="/enwiki/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick</a></li>
<li><a href="/enwiki/wiki/Random_matrix" title="Random matrix">Random</a></li>
<li><a href="/enwiki/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a></li>
<li><a href="/enwiki/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li>
<li><a href="/enwiki/wiki/Shear_matrix" title="Shear matrix">Shear</a></li>
<li><a href="/enwiki/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li>
<li><a href="/enwiki/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li>
<li><a href="/enwiki/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li>
<li><a href="/enwiki/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/enwiki/wiki/Statistics" title="Statistics">statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Centering_matrix" title="Centering matrix">Centering</a></li>
<li><a href="/enwiki/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li>
<li><a href="/enwiki/wiki/Covariance_matrix" title="Covariance matrix">Covariance</a></li>
<li><a href="/enwiki/wiki/Design_matrix" title="Design matrix">Design</a></li>
<li><a href="/enwiki/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li>
<li><a href="/enwiki/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li>
<li><a href="/enwiki/wiki/Projection_matrix" title="Projection matrix">Hat</a></li>
<li><a href="/enwiki/wiki/Precision_(statistics)" title="Precision (statistics)">Precision</a></li>
<li><a href="/enwiki/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li>
<li><a href="/enwiki/wiki/Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/enwiki/wiki/Graph_theory" title="Graph theory">graph theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li>
<li><a href="/enwiki/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li>
<li><a href="/enwiki/wiki/Degree_matrix" title="Degree matrix">Degree</a></li>
<li><a href="/enwiki/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li>
<li><a href="/enwiki/wiki/Incidence_matrix" title="Incidence matrix">Incidence</a></li>
<li><a href="/enwiki/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li>
<li><a href="/enwiki/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li>
<li><a href="/enwiki/wiki/Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa</a></li>
<li><a href="/enwiki/wiki/Density_matrix" title="Density matrix">Density</a></li>
<li><a href="/enwiki/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li>
<li><a href="/enwiki/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li>
<li><a href="/enwiki/wiki/Gamma_matrices" title="Gamma matrices">Gamma</a></li>
<li><a href="/enwiki/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann</a></li>
<li><a href="/enwiki/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li>
<li><a href="/enwiki/wiki/Irregular_matrix" title="Irregular matrix">Irregular</a></li>
<li><a href="/enwiki/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li>
<li><a href="/enwiki/wiki/S-matrix" title="S-matrix">S</a></li>
<li><a href="/enwiki/wiki/State-transition_matrix" title="State-transition matrix">State transition</a></li>
<li><a href="/enwiki/wiki/Substitution_matrix" title="Substitution matrix">Substitution</a></li>
<li><a href="/enwiki/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li>
<li><a href="/enwiki/wiki/Linear_independence" title="Linear independence">Linear independence</a></li>
<li><a href="/enwiki/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li>
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<li><a href="/enwiki/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li>
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