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}.

Any n×n matrix A of the form

$A = \begin{bmatrix} a_{0} & a_{1} & a_{2} & \ldots & \ldots &a_{n-1} \\ a_{1} & a_2 & & & &\vdots \\ a_{2} & & & & & \vdots \\ \vdots & & & & & a_{2n-4}\\ \vdots & & & & a_{2n-4}& a_{2n-3} \\ a_{n-1} & \ldots & \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2} \end{bmatrix}$

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

A_{i,j} = A_{i+1,j-1} = a_{i+j-2}.\

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 $(A_{i,j})_{i,j \ge 1}$ , 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 $\{h_n\}_{n\ge 0}$ is the Hankel transform of the sequence $\{b_n\}_{n\ge 0}$ when

h_n = \det (b_{i+j-2})_{1 \le i,j \le 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 \le i,j \le n+1} = \det (c_{i+j-2})_{1 \le i,j \le n+1}.

Hankel matrices for system identification

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.

Orthogonal polynomials on the real line

Positive Hankel matrices and the Hamburger moment problems

Further information: Hamburger moment problem

Relation between Hankel and Toeplitz matrices

Let $J_n$ be the reflection matrix of order $n$ . For example the reflection matrix of order $5$ is as follows: $J_5 = \begin{bmatrix} & & & & 1 \\ & & & 1 & \\ & & 1 & & \\ & 1 & & & \\ 1 & & & & \\ \end{bmatrix}.$

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.

Notes

References

Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editors—T. Kailath, A.H. Sayed), ch.4 (SIAM).
Victor Y. Pan (2001). Structured matrices and polynomials: unified superfast algorithms. Birkhäuser. ISBN 0817642404.
J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts 13. Cambridge University Press. ISBN 0-521-36791-3.

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