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 matrix of the form

In terms of the components, if the element of is denoted with , and assuming , then we have for all

Properties edit

  • Any Hankel matrix is symmetric.
  • Let   be the   exchange matrix. If   is an   Hankel matrix, then   where   is an   Toeplitz matrix.
    • If   is real symmetric, then   will have the same eigenvalues as   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 edit

Given a formal Laurent series

 
the corresponding Hankel operator is defined as[2]

 
This takes a polynomial   and sends it to the product  , but discards all powers of   with a non-negative exponent, so as to give an element in  , the formal power series with strictly negative exponents. The map   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   is a rational function; that is, a fraction of two polynomials

 

Approximations edit

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   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 edit

The Hankel matrix transform, or simply Hankel transform, of a sequence   is the sequence of the determinants of the Hankel matrices formed from  . Given an integer  , define the corresponding  –dimensional Hankel matrix   as having the matrix elements   Then, the sequence   given by

 

is the Hankel transform 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 edit

Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or hidden Markov model is desired.[3] 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.[4] 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 edit

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.[5]

Positive Hankel matrices and the Hamburger moment problems edit

See also edit

Notes edit

  1. ^ Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835.
  2. ^ Fuhrmann 2012, §8.3
  3. ^ Aoki, Masanao (1983). "Prediction of Time Series". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 38–47. ISBN 0-387-12696-1.
  4. ^ Aoki, Masanao (1983). "Rank determination of Hankel matrices". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 67–68. ISBN 0-387-12696-1.
  5. ^ 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

References edit