Hankel transform of a series

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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.:

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

In mathematical terms:

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

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

1 Hankel transform
2 Hankel matrices for system identification
3 Orthogonal polynomials on the real line
4 See also
5 References

[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

$h_n = \det (b_{i+j})_{1 \le i,j \le n}.$

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

$c_n = \sum_{k=1}^n {n \choose k} b_k$

as the binomial transform of the sequence ${b n}$ , then one has

$\det (b_{i+j})_{1 \le i,j \le n} = \det (c_{i+j})_{1 \le i,j \le n}.$

[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.