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.:

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

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

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

In terms of the components, if the $i,j$ 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 $k=0,...,j-i$.

The Hankel matrix is a symmetric matrix.

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.

A Hankel operator on a Hilbert space is one whose matrix with respect to an orthonormal basis is a (possibly infinite) Hankel matrix $(A_{i,j})_{i,j\geq 1}$, where $A_{i,j}$ depends only on $i+j$.

The determinant of a Hankel matrix is called a catalecticant.

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

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

Here, $a_{i,j}=b_{i+j-2}$ is the Hankel matrix of the sequence $\{b_{n}\}$. 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

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

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. 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. The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.

Orthogonal polynomials on the real line

Positive Hankel matrices and the Hamburger moment problems

Further information: Hamburger moment problem

Relation between Hankel and Toeplitz matrices

Let $J_{n}$ be an exchange matrix of order $n$. If $H(m,n)$ is a $m\times n$ Hankel matrix, then $H(m,n)=T(m,n)\,J_{n}$, where $T(m,n)$ is a $m\times n$ Toeplitz matrix.

If $T(n,n)$ is real symmetric, then $H(n,n)=T(n,n)\,J_{n}$ will have the same eigenvalues as $T(n,n)$ up to sign.

See also

• Cauchy matrix
• Vandermonde matrix
• Displacement rank

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. Vol. 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.