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.;

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 , where a_i,j depends only on i + j.

Contents 1 Hankel transform 2 Hankel matrices for system identification 3 Orthogonal polynomials on the real line 3.1 Positive Hankel matrices and the Hamburger moment problem 3.2 Orthogonal polynomials on the real line 3.3 Tridiagonal model of positive Hankel operators 4 See also

[edit] 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

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

as the binomial transform of the sequence {b_n}, then one has

[edit] 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.

[edit] Orthogonal polynomials on the real line

[edit] Positive Hankel matrices and the Hamburger moment problem

[edit] Orthogonal polynomials on the real line

[edit] Tridiagonal model of positive Hankel operators

[edit] See also

• Hamburger moment problem