# Hankel matrix

Jump to navigation Jump to search

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.:

$\qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.$ 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 $k=0,...,j-i.$ ## Properties

• The Hankel matrix is a symmetric matrix.
• Let $J_{n}$ be the $n\times n$ exchange matrix. If $H$ is a $m\times n$ Hankel matrix, then $H=TJ_{n}$ where $T$ is a $m\times n$ Toeplitz matrix.
• If $T$ is real symmetric, then $H=TJ_{n}$ will have the same eigenvalues as $T$ up to sign.
• 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 $A$ must satisfy, for all rows $i$ and columns $j$ , $(A_{i,j})_{i,j\geq 1}$ . Note that every entry $A_{i,j}$ depends only on $i+j$ .

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

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

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

The determinant of a Hankel matrix is called a catalecticant.

## Hankel matrix transform

The Hankel matrix transform, or simply Hankel transform, produces the sequence of the determinants of the Hankel matrices formed from the given sequence. Namely, the sequence $\{h_{n}\}_{n\geq 0}$ is the Hankel transform of the sequence $\{b_{n}\}_{n\geq 0}$ when

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

### 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.