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In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a n x m matrix in which each ascending skew-diagonal from left to right is constant. For example,

$\qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.$

More generally, a Hankel matrix is any $n\times n$ matrix $A$ of the form

$A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.$

In terms of the components, if the $i,j$ element of $A$ is denoted with $A_{ij}$ , and assuming $i\leq j$ , then we have $A_{i,j}=A_{i+k,j-k}$ for all $k=0,...,j-i.$

Properties

Any Hankel matrix is symmetric.
Let $J_{n}$ $J_{n}$ be the $n\times n$ $n\times n$ exchange matrix. If $H$ $H$ is an $m\times n$ $m\times n$ Hankel matrix, then $H=TJ_{n}$ $H=TJ_{n}$ where $T$ $T$ is an $m\times n$ $m\times n$ Toeplitz matrix.
- If $T$ is real symmetric, then $H=TJ_{n}$ will have the same eigenvalues as $T$ up to sign.^[1]
The Hilbert matrix is an example of a Hankel matrix.
The determinant of a Hankel matrix is called a catalecticant.

Hankel operator

Given a formal Laurent series $f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},$ the corresponding Hankel operator is defined as^[2] $H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].$ This takes a polynomial $g\in \mathbf {C} [z]$ and sends it to the product $fg$ , but discards all powers of $z$ with a non-negative exponent, so as to give an element in $z^{-1}\mathbf {C} [[z^{-1}]]$ , the formal power series with strictly negative exponents. The map $H_{f}$ is in a natural way $\mathbf {C} [z]$ -linear, and its matrix with respect to the elements $1,z,z^{2},\dots \in \mathbf {C} [z]$ and $z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]$ is the Hankel matrix ${\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.$ Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if $f$ is a rational function, that is, a fraction of two polynomials $f(z)={\frac {p(z)}{q(z)}}.$

Approximations

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 $A$ 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.

Hankel matrix transform

The Hankel matrix transform, or simply Hankel transform, of a sequence $b_{k}$ is the sequence of the determinants of the Hankel matrices formed from $b_{k}$ . Given an integer $n>0$ , define the corresponding $(n\times n)$ -dimensional Hankel matrix $B_{n}$ as having the matrix elements $[B_{n}]_{i,j}=b_{i+j}.$ Then the sequence $h_{n}$ given by $h_{n}=\det B_{n}$ is the Hankel transform of the sequence $b_{k}.$ The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes $c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}$ as the binomial transform of the sequence $b_{n}$ , then one has $\det B_{n}=\det C_{n}.$

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.^[3] 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.^[4] 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 applied to polynomial distributions results in a Hankel matrix that needs to be inverted in order to obtain the weight parameters of the polynomial distribution approximation.^[5]

Positive Hankel matrices and the Hamburger moment problems

Notes

^ Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835.
^ Fuhrmann 2012, §8.3
^ Aoki, Masanao (1983). "Prediction of Time Series". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 38–47. ISBN 0-387-12696-1.
^ Aoki, Masanao (1983). "Rank determination of Hankel matrices". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 67–68. ISBN 0-387-12696-1.
^ 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

References

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 (SIAM).
Fuhrmann, Paul A. (2012). A polynomial approach to linear algebra. Universitext (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4614-0338-8. ISBN 978-1-4614-0337-1. Zbl 1239.15001.

Victor Y. Pan (2001). Structured matrices and polynomials: unified superfast algorithms. Birkhäuser. ISBN 0817642404.
J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts. Vol. 13. Cambridge University Press. ISBN 0-521-36791-3.

[simax1-1] Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835.

[2] Fuhrmann 2012, §8.3

[3] Aoki, Masanao (1983). "Prediction of Time Series". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 38–47. ISBN 0-387-12696-1.

[4] Aoki, Masanao (1983). "Rank determination of Hankel matrices". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 67–68. ISBN 0-387-12696-1.

[PolyD2-5] 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

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[2]

[3]

[4]

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@@ Line 1: / Line 1: @@
-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.:
+{{Short description|A square matrix in which each ascending skew-diagonal from left to right is constant}}
+In [[linear algebra]], a '''Hankel matrix''' (or '''[[catalecticant]] matrix'''), named after [[Hermann Hankel]], is a n x m matrix in which each ascending skew-diagonal from left to right is constant. For example,
-:<math>\begin{bmatrix}
+<math display=block>\qquad\begin{bmatrix}
 a & b & c & d & e \\
 b & c & d & e & f \\
@@ Line 9: / Line 10: @@
 \end{bmatrix}.</math>
-More generally, a '''Hankel matrix''' is any <math>n\times n</math> matrix <math>A</math> of the form
+More generally, a '''Hankel matrix''' is any <math>n \times n</math> [[matrix (mathematics)|matrix]] <math>A</math> of the form
+<math display=block>A = \begin{bmatrix}
-<math>
+  a_0 & a_1 & a_2 & \ldots & a_{n-1}  \\
-A =
+  a_1 & a_2 &  &  &\vdots \\
-\begin{bmatrix}
-  a_{0} & a_{1} & a_{2} & \ldots & \ldots  &a_{n-1}  \\
+  a_2 &  &  & & a_{2n-4} \\
-  a_{1} & a_2 &  &  & &\vdots \\
+ \vdots & &  & a_{2n-4} & a_{2n-3} \\
-  a_{2} &  &  & & & \vdots \\
+a_{n-1} &  \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2}
+\end{bmatrix}.</math>
- \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 display="block">A_{i,j} = A_{i+k,j-k}</math>for all <math>k=0,...,j-i</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]].
+* 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 an <math>m \times n</math> Hankel matrix, then <math>H = T J_n</math> where <math>T</math> is an <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.
+* The [[determinant]] of a Hankel matrix is called a [[catalecticant]].
+==Hankel operator==
-The Hankel matrix is closely related to the [[Toeplitz matrix]] (a Hankel matrix is an upside-down Toeplitz matrix). For a special case of this matrix see [[Hilbert matrix]].
+Given a [[formal Laurent series]]
+<math display="block">
+ f(z) = \sum_{n=-\infty}^N a_n z^n,
+</math>
+the corresponding '''Hankel operator''' is defined as<ref>{{harvnb|Fuhrmann|2012|loc=§8.3}}</ref>
+<math display="block">
+ H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf C[[z^{-1}]].
+</math>
+This takes a [[polynomial]] <math>g \in \mathbf C[z]</math> and sends it to the product <math>fg</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 & \vdots & \ddots
+\end{bmatrix}.</math>
+Any Hankel matrix arises in this 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]], that is, a fraction of two polynomials
+<math display="block">
+ f(z) = \frac{p(z)}{q(z)}.
+</math>
+==Approximations==
-A Hankel [[operator (mathematics)|operator]] on a [[Hilbert space]] is one whose matrix with respect to an [[orthonormal basis]] is a (possibly infinite) Hankel matrix
+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.
-<math>(A_{i,j})_{i,j \ge 1}</math>, where <math> A_{i,j}</math> depends only on <math>i+j</math>.
+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 transform==
+==Hankel matrix transform==
+{{Distinguish|Hankel transform}}
-The '''Hankel transform''' is the name sometimes given to the transformation of a [[sequence]], where the transformed sequence corresponds to the determinant of the Hankel matrix.  That is, the sequence <math>\{h_n\}_{n\ge 0}</math> is the Hankel transform of the sequence <math>\{b_n\}_{n\ge 0}</math> when
+The '''Hankel matrix transform''', or simply '''Hankel transform''', of a [[sequence]] <math>b_k</math> is the sequence of the determinants of the Hankel matrices formed from <math>b_k</math>. Given an integer <math>n > 0</math>, define the corresponding <math>(n \times n)</math>-dimensional Hankel matrix <math>B_n</math> as having the matrix elements <math>[B_n]_{i,j} = b_{i+j}.</math> Then the sequence <math>h_n</math> given by
-:<math>h_n = \det (b_{i+j-2})_{1 \le i,j \le n+1}.</math>
+<math display="block">
+ h_n = \det B_n
-Here, <math>a_{i,j}=b_{i+j-2}</math> is the Hankel matrix of the sequence <math>\{b_n\}</math>.  The Hankel transform is invariant under the [[binomial transform]] of a sequence.  That is, if one writes
+</math>
+is the Hankel transform of the sequence <math>b_k.</math> The Hankel transform is invariant under the [[binomial transform]] of a sequence. That is, if one writes
-:<math>c_n = \sum_{k=0}^n {n \choose k} b_k</math>
+<math display="block">
+ c_n = \sum_{k=0}^n {n \choose k} b_k
-as the binomial transform of the sequence <math>\{b_n\}</math>, then one has
+</math>
+as the binomial transform of the sequence <math>b_n</math>, then one has <math>\det B_n = \det C_n.</math>
-:<math>\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 |authorlink=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 |chapterurl=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 |chapterurl=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.
+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 ===
-== Orthogonal polynomials on the real line ==
+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}}
+{{Further|Hamburger moment problem}}
-=== Orthogonal polynomials on the real line ===
-=== Tridiagonal model of positive Hankel operators ===
-=== Relation between Hankel and Toeplitz matrices ===
-Let <math> J_n </math> be an [[exchange matrix]] of order <math> n </math>.  If <math>H(m,n)</math> is a <math>m \times n </math> Hankel matrix, then <math>H(m,n) = T(m, n) \, J_n </math>, where <math>T(m,n)</math> is a <math>m \times n </math> Toeplitz matrix.
-If <math>T(n,n)</math> is symmetric, then <math>H(n,n) = T(n, n) \, J_n </math> will have the same eigenvalues as <math>T(n,n)</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>
-=== Relations between structured matrices ===
 ==See also==
 * [[Cauchy matrix]]
+* [[Jacobi operator]]
+* [[Toeplitz matrix]], an "upside down" (that is, row-reversed) Hankel matrix
 * [[Vandermonde matrix]]
-* [[Displacement rank]]
 == Notes ==
-{{reflist}}
+{{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&mdash;T. Kailath, A.H. Sayed), ch.4 ([[Society for Industrial and Applied Mathematics|SIAM]]).
+*{{cite book
-* {{cite book | title=Structured matrices and polynomials: unified superfast algorithms | author=Victor Y. Pan | authorlink=Victor Pan  | publisher=[[Birkhäuser]] | year=2001 | isbn=0817642404 }}
+ | last = Fuhrmann
-* {{cite book | title=An introduction to Hankel operators | author=J.R. Partington | authorlink=Jonathan Partington | series=LMS Student Texts | volume=13 | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-36791-3 }}
+ | first = Paul A.
-*  P. Jain and R.B. Pachori, [https://www.sciencedirect.com/science/article/pii/S0016003215002288 An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix], Journal of the Franklin Institute, vol. 352, issue 10, pp. 4017--4044, October 2015.
+ | title = A polynomial approach to linear algebra
-*  P. Jain and R.B. Pachori, [https://ieeexplore.ieee.org/document/6847702/ Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of Hankel matrix], IEEE/ACM Transactions on Audio, Speech and Language Processing, vol. 22. issue 10, pp. 1467-1482, October 2014.
+ | edition = 2
-*R.R. Sharma and R.B. Pachori, [https://ieeexplore.ieee.org/document/8249682/ Time-frequency representation using IEVDHM-HT with application to classification of epileptic EEG signals], IET Science, Measurement & Technology, vol. 12, issue 01, pp. 72-82, January 2018.
+ | 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 = 1239.15001
+}}
+* {{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}}
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