# Hankel matrix

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example,
$\qquad {\begin{bmatrix}a&b&c&d&e\\b&c&d&e&f\\c&d&e&f&g\\d&e&f&g&h\\e&f&g&h&i\\\end{bmatrix}}.$
More generally, a Hankel matrix is any $n\times n$ matrix $A$ of the form
$A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.$
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

* Any Hankel matrix is symmetric.
* Let $J_{n}$ be the $n\times n$ exchange matrix. If $H$ is an $m\times n$ Hankel matrix, then $H=TJ_{n}$ where $T$ is an $m\times n$ 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.
* The determinant of a Hankel matrix is called a catalecticant.

## Hankel operator

Given a formal Laurent series $f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},$ the corresponding Hankel operator is defined as $H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].$ This takes a polynomial $g\in \mathbf {C} [z]$ 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 $z^{-1}\mathbf {C} [[z^{-1}]]$, the formal power series with strictly negative exponents. The map $H_{f}$ is in a natural way $\mathbf {C} [z]$-linear, and its matrix with respect to the elements $1,z,z^{2},\dots \in \mathbf {C} [z]$ and $z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]$ is the Hankel matrix ${\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.$ 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 $f(z)={\frac {p(z)}{q(z)}}.$

## Approximations

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.

## Hankel matrix 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 $(n\times n)$-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 $\det B_{n}=\det C_{n}.$

## 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.