Hankel matrix

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.:

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

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

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

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

Some properties and facts

• The Hankel matrix is a symmetric matrix.
• Let ${\displaystyle J_{n}}$ be an exchange matrix of order ${\displaystyle n}$. If ${\displaystyle H(m,n)}$ is a ${\displaystyle m\times n}$ Hankel matrix, then ${\displaystyle H(m,n)=T(m,n)\,J_{n}}$, where ${\displaystyle T(m,n)}$ is a ${\displaystyle m\times n}$ Toeplitz matrix.
  • If ${\displaystyle T(n,n)}$ is real symmetric, then ${\displaystyle H(n,n)=T(n,n)\,J_{n}}$ will have the same eigenvalues as ${\displaystyle T(n,n)}$ up to sign.[1]

• The Hilbert matrix is an example of a Hankel matrix.

Hankel operator

A Hankel 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 ${\displaystyle A}$ must satisfy, for all rows ${\displaystyle i}$ and columns ${\displaystyle j}$, ${\displaystyle (A_{i,j})_{i,j\geq 1}}$. Note that every entry ${\displaystyle A_{i,j}}$ depends only on ${\displaystyle i+j}$.

Let the corresponding Hankel Operator be ${\displaystyle H_{\alpha }}$. Given a Hankel matrix ${\displaystyle A}$, the corresponding Hankel operator is then defined as ${\displaystyle H_{\alpha }(u)=Au}$.

We are often interested in Hankel operators ${\displaystyle H_{\alpha }:\ell ^{2}\left(Z^{+}\cup \{0\}\right)\rightarrow \ell ^{2}\left(\mathbb {Z} ^{+}\cup \{0\}\right)}$ over the Hilbert space ${\displaystyle \ell ^{2}(\mathbf {Z} )}$, the space of square integrable bilateral complex sequences. For any ${\displaystyle u\in \ell ^{2}(\mathbf {Z} )}$, we have

${\displaystyle \|u\|_{\ell ^{2}(z)}^{2}=\sum _{n=-\infty }^{\infty }\left|u_{n}\right|^{2}}$

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 matrix ${\displaystyle A}$ 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.

The determinant of a Hankel matrix is called a catalecticant.

Hankel transform

See also: 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 ${\displaystyle \{h_{n}\}_{n\geq 0}}$ is the Hankel transform of the sequence ${\displaystyle \{b_{n}\}_{n\geq 0}}$ when

${\displaystyle h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.}$

Here, ${\displaystyle a_{i,j}=b_{i+j-2}}$ is the Hankel matrix of the sequence ${\displaystyle \{b_{n}\}}$. The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes

${\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}$

as the binomial transform of the sequence ${\displaystyle \{b_{n}\}}$, then one has

${\displaystyle \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.}$

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.[2] 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.[3] 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 [4].

Positive Hankel matrices and the Hamburger moment problems

Further information: Hamburger moment problem

See also

• Cauchy matrix
• Vandermonde matrix

Notes

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. 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.
3. 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.
4. 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).
• 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. 13. Cambridge University Press. ISBN 0-521-36791-3.
• P. Jain and R.B. Pachori, 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.
• P. Jain and R.B. Pachori, 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.
• R.R. Sharma and R.B. Pachori, 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.