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.:

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

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

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

In terms of the components, if the ${\displaystyle i,j}$ element of ${\displaystyle A}$ is denoted with ${\displaystyle A_{ij}}$, and assuming ${\displaystyle i\leq j}$, then we have ${\displaystyle A_{i,j}=A_{i+k,j-k}}$for all ${\displaystyle k=0,...,j-i}$.

Some properties and facts

• The Hankel matrix is a symmetric matrix.
• Let ${\displaystyle J_{n}}$  be an exchange matrix of order ${\displaystyle n}$ . 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.
• If ${\displaystyle T(n,n)}$  is real symmetric, then ${\displaystyle H(n,n)=T(n,n)\,J_{n}}$  will have the same eigenvalues as ${\displaystyle T(n,n)}$  up to sign.[1]

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 ${\displaystyle (A_{i,j})_{i,j\geq 1}}$ , 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.[2] 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.[3] The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.