Hankel matrix

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

:

In mathematical terms:

:$a\_\{i,j\}\; =\; a\_\{i-1,j+1\}.,$

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 $(a\_\{i,j\})\_\{i,j\; ge\; 0\}$, where $a\_\{i,j\}$ depends only on $i+j$.

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 $\{h\_n\}$ is the Hankel transform of the sequence $\{b\_n\}$ when

:$h\_n\; =\; det\; (b\_\{i+j\})\_\{0\; le\; i,j\; le\; n\}.$

Here, $a\_\{i,j\}=b\_\{i+j\}$ 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

:$det\; (b\_\{i+j\})\_\{0\; le\; i,j\; le\; n\}\; =\; det\; (c\_\{i+j\})\_\{0\; le\; i,j\; le\; n\}.$

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 problem

Orthogonal polynomials on the real line

Tridiagonal model of positive Hankel operators

ee also

* Hamburger moment problem

References

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