Hankel matrix: Difference between revisions

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.:

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

In mathematical terms:

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

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 ${\displaystyle (a_{i,j})_{i,j\geq 1}}$, where ${\displaystyle a_{i,j}}$ depends only on ${\displaystyle i+j}$.

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 ${\displaystyle \{h_{n}\}_{n\geq 0}}$ is the Hankel transform of the sequence ${\displaystyle \{b_{n}\}_{n\geq 0}}$ when

${\displaystyle h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.}$

Here, ${\displaystyle a_{i,j}=b_{i+j-2}}$ is the Hankel matrix of the sequence ${\displaystyle \{b_{n}\}}$. The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes

${\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}$

as the binomial transform of the sequence ${\displaystyle \{b_{n}\}}$, then one has

${\displaystyle \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.}$

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

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Positive Hankel matrices and the Hamburger moment problems

Orthogonal polynomials on the real line

Tridiagonal model of positive Hankel operators

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. Vol. 13. Cambridge University Press. ISBN 0-521-36791-3.

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