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]- Cauchy matrix
- Jacobi operator
- Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix
- Vandermonde matrix
Notes
[edit]- ^ 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.
- ^ Fuhrmann 2012, §8.3
- ^ 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.
- ^ 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.
- ^ 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]- 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).
- Fuhrmann, Paul A. (2012). A polynomial approach to linear algebra. Universitext (2 ed.). New York, NY: Springer. doi:10.1007/978-1-4614-0338-8. ISBN 978-1-4614-0337-1. Zbl 1239.15001.
- Victor Y. Pan (2001). Structured matrices and polynomials: unified superfast algorithms. Birkhäuser. ISBN 0817642404.
- J.R. Partington (1988). An introduction to Hankel operators. LMS Student Texts. Vol. 13. Cambridge University Press. ISBN 0-521-36791-3.