Hankel matrix explained

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant. For example,

More generally, a Hankel matrix is any

n x 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 A _{i,j} = A _{i+k,j-k} for all k = 0 ,...,j - i.

Properties

• Any Hankel matrix is symmetric.
• Let

J _{n} be the n x n exchange matrix. If H is an m x n Hankel matrix, then H = T J _{n} where T is an m x n Toeplitz matrix.

• • If

T is real symmetric, then H = T J _{n} will have the same eigenvalues as T up to sign.^[1]

• The Hilbert matrix is an example of a Hankel matrix.
• The determinant of a Hankel matrix is called a catalecticant.

Hankel operator

g \in C[z] 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 z ^{-1} C[[z ^{-1} ]] , the formal power series with strictly negative exponents. The map H _{f} is in a natural way C[z] -linear, and its matrix with respect to the elements 1 , z, z ^{2,} ... \in C[z] and z ^{-1} , z ^{-2} , ... \in z ^{-1} C[[z ^{-1} ]] 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$$f(z)\; =\; \backslash frac.$$

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

b _{k} is the sequence of the determinants of the Hankel matrices formed from b _{k} . Given an integer n > 0 , define the corresponding (n x n) -dimensional Hankel matrix B _{n} as having the matrix elements [B _{n]} _{i,j} = b _{i+j} . Then the sequence h _{n} given by $$h\_n\; =\; \backslash det\; B\_n$$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$$c\_n\; =\; \backslash sum\_^n\; b\_k$$as the binomial transform of the sequence b _{n} , then one has \det B _{n} = \det C _{n.}

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.^[2] 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.^[3] 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.^[4]

Positive Hankel matrices and the Hamburger moment problems

See also

• Cauchy matrix
• Jacobi operator
• Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix
• Vandermonde matrix

References

• Brent R.P. (1999), "Stability of fast algorithms for structured linear systems", Fast Reliable Algorithms for Matrices with Structure (editors - T. Kailath, A.H. Sayed), ch.4 (SIAM).
• Book: Fuhrmann , Paul A. . A polynomial approach to linear algebra . 2 . Universitext . 2012 . Springer . New York, NY . 978-1-4614-0337-1 . 10.1007/978-1-4614-0338-8 . 1239.15001.
• Book: Structured matrices and polynomials: unified superfast algorithms . Victor Y. Pan . Victor Pan . . 2001 . 0817642404 .
• Book: An introduction to Hankel operators . J.R. Partington . Jonathan Partington . LMS Student Texts . 13 . . 1988 . 0-521-36791-3 .

Notes and References

1. Yasuda . M. . A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices . SIAM J. Matrix Anal. Appl. . 25 . 3 . 601–605 . 2003 . 10.1137/S0895479802418835.
2. Book: Aoki, Masanao . Masanao Aoki . Prediction of Time Series . Notes on Economic Time Series Analysis : System Theoretic Perspectives . New York . Springer . 1983 . 0-387-12696-1 . 38–47 . https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA38 .
3. Book: Aoki, Masanao . Rank determination of Hankel matrices . Notes on Economic Time Series Analysis : System Theoretic Perspectives . New York . Springer . 1983 . 0-387-12696-1 . 67–68 . https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA67 .
4. J. Munkhammar, L. Mattsson, J. Rydén (2017) "Polynomial probability distribution estimation using the method of moments". PLoS ONE 12(4): e0174573. https://doi.org/10.1371/journal.pone.0174573