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%2B1}.\$

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 $i%2Bj$ .

The determinant of a Hankel matrix is called a catalecticant.

1 Hankel transform
2 Hankel matrices for system identification
3 Orthogonal polynomials on the real line
4 See also
5 References

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

$h_n = \det (b_{i%2Bj-2})_{1 \le i,j \le n%2B1}.$

Here, $a_{i,j}=b_{i%2Bj-2}$ 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

$\det (b_{i%2Bj-2})_{1 \le i,j \le n%2B1} = \det (c_{i%2Bj-2})_{1 \le i,j \le n%2B1}.$

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

Orthogonal polynomials on the real line

Tridiagonal model of positive Hankel operators

References

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

Hankel matrix

Contents

Hankel transform

Hankel matrices for system identification

Orthogonal polynomials on the real line

Positive Hankel matrices and the Hamburger moment problems

Orthogonal polynomials on the real line

Tridiagonal model of positive Hankel operators

See also

References