In linear algebra, a Hankel matrix, named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: More generally, a Hankel matrix is any matrix of the form In terms of the components, if the element of is denoted with, and assuming, then we have for all.
A Hankel operator on a Hilbert space is one whose matrix is a Hankel matrix, with respect to an orthonormal basis. As indicated above, a Hankel Matrix is a matrix with constant values along its antidiagonals, which means that a Hankel matrix must satisfy, for all rows and columns,. Note that every entry depends only on. Let the corresponding Hankel Operator be. Given a Hankel matrix, the corresponding Hankel operator is then defined as. We are often interested in Hankel operators over the Hilbert space, the space of square integrable bilateral complex sequences. For any, we have 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 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 matrix 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. The determinant of a Hankel matrix is called a catalecticant.
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 Here, is the Hankel matrix of the sequence. The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes as the binomial transform of the sequence, 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.
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
