Hankel matrix

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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 Ai,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 a (possibly infinite) Hankel matrix (A_{{i,j}})_{{i,j\geq 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\geq 0}} is the Hankel transform of the sequence \{b_{n}\}_{{n\geq 0}} when

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

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

See also

References


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