# Hankel matrix

(Redirected from Hankel transform of a series)

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.:

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

Any n×n matrix A of the form

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

is a Hankel matrix. If the i,j element of A is denoted Ai,j, then we have

${\displaystyle A_{i,j}=A_{j,i}=a_{i+j-2}.\ }$

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 ${\displaystyle (A_{i,j})_{i,j\geq 1}}$, where ${\displaystyle A_{i,j}}$ depends only on ${\displaystyle i+j}$.

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 ${\displaystyle \{h_{n}\}_{n\geq 0}}$ 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. 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

### Relation between Hankel and Toeplitz matrices

Let ${\displaystyle J_{n}}$ be the reflection matrix of order ${\displaystyle n}$. For example the reflection matrix of order ${\displaystyle 5}$ is as follows: ${\displaystyle J_{5}={\begin{bmatrix}&&&&1\\&&&1&\\&&1&&\\&1&&&\\1&&&&\\\end{bmatrix}}.}$

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.