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.;

<math>\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}</math>

In mathematical terms:

<math>a_{i,j} = a_{i-1,j+1}</math>

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 <math>(a_{i,j})_{i,j \ge 0}</math>, where <math> a_{i,j}</math> depends only on <math>i+j</math>.

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 <math>\{h_n\}</math> is the Hankel transform of the sequence <math>\{b_n\}</math> when

<math>h_n = \det (b_{i+j})_{0 \le i,j \le n}</math>

Here, <math>a_{i,j}=b_{i+j}</math> is the Hankel matrix of the sequence <math>\{b_n\}</math>. The Hankel transform of a sequence commutes with the binomial transform of a sequence. That is, if one writes

<math>c_n = \sum_{k=0}^n {n \choose k} b_k</math>

as the binomial transform of the sequence <math>\{b_n\}</math>, then one has

<math>\det (b_{i+j})_{0 \le i,j \le n} = \det (c_{i+j})_{0 \le i,j \le n}</math>