In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a n x m matrix in which each ascending skew-diagonal from left to right is constant. For example, More generally, a Hankel matrix is any n \times n matrix A of the form In terms of the components, if the i,j element of A is denoted with A_{ij}, and assuming i \le j, then we have for all

Properties

Hankel operator

Given a formal Laurent series the corresponding Hankel operator is defined as This takes a polynomial and sends it to the product fg, but discards all powers of z with a non-negative exponent, so as to give an element in, the formal power series with strictly negative exponents. The map H_f is in a natural way -linear, and its matrix with respect to the elements and is the Hankel matrix Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if f is a rational function, that is, a fraction of two polynomials

Approximations

We are often interested in approximations of the Hankel operators, possibly by low-order operators. In order to approximate the output of the operator, we can use the spectral norm (operator 2-norm) to measure the error of our approximation. This suggests singular value decomposition as a possible technique to approximate the action of the operator. Note that the matrix A does not have to be finite. If it is infinite, traditional methods of computing individual singular vectors will not work directly. We also require that the approximation is a Hankel matrix, which can be shown with AAK theory.

Hankel matrix transform

The Hankel matrix transform, or simply Hankel transform, of a sequence b_k is the sequence of the determinants of the Hankel matrices formed from b_k. Given an integer n > 0, define the corresponding -dimensional Hankel matrix B_n as having the matrix elements Then the sequence h_n given by is the Hankel transform of the sequence b_k. 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

Applications of Hankel matrices

Hankel matrices are formed when, given a sequence of output data, 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. The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.

Method of moments for polynomial distributions

The method of moments applied to polynomial distributions results in a Hankel matrix that needs to be inverted in order to obtain the weight parameters of the polynomial distribution approximation.

Positive Hankel matrices and the Hamburger moment problems