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 .

Some properties and facts[edit]

  • The Hankel matrix is a symmetric matrix.
  • Let be an exchange matrix of order . If is a Hankel matrix, then , where is a Toeplitz matrix.
    • If is real symmetric, then will have the same eigenvalues as up to sign.[1]

Hankel operator[edit]

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.[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.

Positive Hankel matrices and the Hamburger moment problems[edit]

See also[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. ^ 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.
  3. ^ 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.