Hankel matrix

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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, 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 .

The Hankel matrix is a symmetric matrix.

The Hankel matrix is closely related to the Toeplitz matrix (a Hankel matrix is an upside-down Toeplitz matrix). For a special case of this matrix see Hilbert matrix.

A Hankel operator on a Hilbert space is one whose matrix with respect to an orthonormal basis is a (possibly infinite) Hankel matrix , where depends only on .

The determinant of a Hankel matrix is called a catalecticant.

Hankel transform[edit]

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[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. 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.

Orthogonal polynomials on the real line[edit]

Positive Hankel matrices and the Hamburger moment problems[edit]

Orthogonal polynomials on the real line[edit]

Tridiagonal model of positive Hankel operators[edit]

Relation between Hankel and Toeplitz matrices[edit]

Let be the reflection matrix of order . For example the reflection matrix of order is as follows:

If is a Hankel matrix, then , where is a Toeplitz matrix.

Relations between structured matrices[edit]

See also[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).
  • 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. 13. Cambridge University Press. ISBN 0-521-36791-3.
  • P. Jain and R.B. Pachori, An iterative approach for decomposition of multi-component non-stationary signals based on eigenvalue decomposition of the Hankel matrix, Journal of the Franklin Institute, vol. 352, issue 10, pp. 4017--4044, October 2015.
  • P. Jain and R.B. Pachori, Event-based method for instantaneous fundamental frequency estimation from voiced speech based on eigenvalue decomposition of Hankel matrix, IEEE/ACM Transactions on Audio, Speech and Language Processing, vol. 22. issue 10, pp. 1467-1482, October 2014.
  • R.R. Sharma and R.B. Pachori, Time-frequency representation using IEVDHM-HT with application to classification of epileptic EEG signals, IET Science, Measurement & Technology, vol. 12, issue 01, pp. 72-82, January 2018.