Hankel matrix

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

[\begin

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]

In mathematical terms:

[a_ = a_]

The Hankel matrix is closely related to the Toeplitz matrix (a Hankel matrix is an upside-down Toeplitz matrix).

A Hankel operator on a Hilbert space is one whose matrix with respect to an orthonormal basis is an infinite Hankel matrix [(a_)_], where [ a_] 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 [\] is the Hankel transform of the sequence [\] when

[h_n = \det (b_)_]

Here, [a_=b_] is the Hankel matrix of the sequence [\]. The Hankel transform of a sequence commutes with the binomial transform of a sequence. That is, if one writes

[c_n = \sum_^n b_k]

as the binomial transform of the sequence [\], then one has

[\det (b_)_ = \det (c_)_]

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

 

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