Hankel matrix

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix with constant skew-diagonals (positive sloping diagonals), e.g.:

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

If the i,j element of A is denoted A_i,j, then we have

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

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 an 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 and 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

See also

• Hamburger moment problem
• Toeplitz matrix, a Hankel matrix 'upside-down'.

References

• J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts. 13. Cambridge University Press. ISBN 0-521-36791-3.