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.

See also

• Cauchy matrix
• Vandermonde matrix
• Displacement rank

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).