# 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 $k=0,...,j-i$ .

## Some properties and facts

• The Hankel matrix is a symmetric matrix.
• 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.

## Hankel operator

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.