Hankel matrix

• Any Hankel matrix is symmetric.
• Let $J\_n$ be the $n\; \backslash times\; n$ exchange matrix. If $H$ is an $m\; \backslash times\; n$ Hankel matrix, then $H\; =\; T\; J\_n$ where $T$ is an $m\; \backslash times\; n$ Toeplitz matrix.
• If $T$ is real symmetric, then $H\; =\; T\; J\_n$ will have the same eigenvalues as $T$ up to sign.{{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}}
• The Hilbert matrix is an example of a Hankel matrix.
• The determinant of a Hankel matrix is called a catalecticant.

Given a formal Laurent series

f(z) = \sum_{n=-\infty}^N a_n z^n,

the corresponding Hankel operator is defined as{{harvnb|Fuhrmann|2012|loc=§8.3}}

H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf Cz^{-1}.

This takes a polynomial $g\; \backslash in\; \backslash mathbf\; C[z]$ 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 $z^\{-1\}\; \backslash mathbf\; Cz^\{-1\}$, the formal power series with strictly negative exponents. The map $H\_f$ is in a natural way $\backslash mathbf\; C[z]$-linear, and its matrix with respect to the elements $1,\; z,\; z^2,\; \backslash dots\; \backslash in\; \backslash mathbf\; C[z]$ and $z^\{-1\},\; z^\{-2\},\; \backslash dots\; \backslash in\; z^\{-1\}\backslash mathbf\; Cz^\{-1\}$ is the Hankel matrix

$$\backslash begin\{bmatrix\}$$

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)}.

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.

{{Distinguish|Hankel 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 $(n\; \backslash times\; n)$-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 $\backslash det\; B\_n\; =\; \backslash det\; C\_n.$

Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired.{{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 }} 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.{{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 }} The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.

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.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

{{Further|Hamburger moment problem}}

• Cauchy matrix
• Jacobi operator
• Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix
• Vandermonde matrix

{{Reflist}}

• 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).
• {{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 = 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}}

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Category:Matrices (mathematics)

Category:Transforms