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A Hankel matrix doesn't have to be a square matrix
 
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{{Short description|A square matrix in which each ascending skew-diagonal from left to right is constant}}
{{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 [[square matrix]] in which each ascending skew-diagonal from left to right is constant. For example,
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 display=block>\qquad\begin{bmatrix}
<math display=block>\qquad\begin{bmatrix}
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==Hankel operator==
==Hankel operator==
Given a [[formal Laurent series]]
Given a [[formal Laurent series]]
<math display="block">f(z) = \sum_{n=-\infty}^N a_n z^n</math>
<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>
the corresponding '''Hankel operator''' is defined as<ref>{{harvnb|Fuhrmann|2012|loc=§8.3}}</ref>
<math display="block">

<math display="block">H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf C[[z^{-1}]],</math>
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
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}
<math display=block>\begin{bmatrix}
a_1 & a_2 & \ldots \\
a_1 & a_2 & \ldots \\
a_2 & a_3 & \ldots \\
a_2 & a_3 & \ldots \\
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\vdots & \vdots & \ddots
\vdots & \vdots & \ddots
\end{bmatrix}.</math>
\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
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">

<math display="block">f(z) = \frac{p(z)}{q(z)}.</math>
f(z) = \frac{p(z)}{q(z)}.
</math>


==Approximations==
==Approximations==
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{{Distinguish|Hankel transform}}
{{Distinguish|Hankel transform}}


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
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 display="block">h_n = \det B_n</math>
<math display="block">
h_n = \det B_n
</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 display="block">
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 display="block">c_n = \sum_{k=0}^n {n \choose k} b_k</math>
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>\det B_n = \det C_n.</math>
as the binomial transform of the sequence <math>b_n</math>, then one has <math>\det B_n = \det C_n.</math>



Latest revision as of 15:26, 29 November 2024

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,

More generally, a Hankel matrix is any matrix of the form

In terms of the components, if the element of is denoted with , and assuming , then we have for all

Properties

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  • Any Hankel matrix is symmetric.
  • Let be the exchange matrix. If is an Hankel matrix, then where is an Toeplitz matrix.
    • If is real symmetric, then will have the same eigenvalues as 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

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Given a formal Laurent series the corresponding Hankel operator is defined as[2] This takes a polynomial and sends it to the product , but discards all powers of with a non-negative exponent, so as to give an element in , the formal power series with strictly negative exponents. The map is in a natural way -linear, and its matrix with respect to the elements and is the Hankel matrix Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if is a rational function, that is, a fraction of two polynomials

Approximations

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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 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

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The Hankel matrix transform, or simply Hankel transform, of a sequence is the sequence of the determinants of the Hankel matrices formed from . Given an integer , define the corresponding -dimensional Hankel matrix as having the matrix elements Then the sequence given by is the Hankel transform of the sequence The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes as the binomial transform of the sequence , then one has

Applications of Hankel matrices

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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

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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

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See also

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Notes

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  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.
  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

References

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