Hankel matrix

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Square matrix in which each ascending skew-diagonal from left to right is constant

title: "Hankel matrix" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["matrices-(mathematics)", "transforms"] description: "Square matrix in which each ascending skew-diagonal from left to right is constant" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Hankel_matrix" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Square matrix in which each ascending skew-diagonal from left to right is constant ::

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

\qquad\begin{bmatrix} a & b & c & d & e \ b & c & d & e & f \ c & d & e & f & g \ d & e & f & g & h \ e & f & g & h & i \ \end{bmatrix}.

More generally, a Hankel matrix is any n \times n matrix A of the form

A = \begin{bmatrix} a_0 & a_1 & a_2 & \ldots & a_{n-1} \ a_1 & a_2 & & &\vdots \ a_2 & & & & a_{2n-4} \ \vdots & & & a_{2n-4} & a_{2n-3} \ a_{n-1} & \ldots & a_{2n-4} & a_{2n-3} & a_{2n-2} \end{bmatrix}.

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

Hankel operator

Given a formal Laurent series f(z) = \sum_{n=-\infty}^N a_n z^n, the corresponding Hankel operator is defined as H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf Cz^{-1}. This takes a polynomial g \in \mathbf 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} \mathbf Cz^{-1}, the formal power series with strictly negative exponents. The map H_f is in a natural way \mathbf C[z]-linear, and its matrix with respect to the elements 1, z, z^2, \dots \in \mathbf C[z] and z^{-1}, z^{-2}, \dots \in z^{-1}\mathbf Cz^{-1} is the Hankel matrix \begin{bmatrix} a_1 & a_2 & \ldots \ a_2 & a_3 & \ldots \ a_3 & a_4 & \ldots \ \vdots & \vdots & \ddots \end{bmatrix}. 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) = \frac{p(z)}{q(z)}.

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

The Hankel matrix transform, or simply Hankel transform, of a sequence 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 \times 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 = \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 = \sum_{k=0}^n {n \choose k} 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. 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.

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.

Positive Hankel matrices and the Hamburger moment problems

Notes

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).

  • {{cite book | last = Fuhrmann | first = Paul A. | title = A polynomial approach to linear algebra | edition = 2 | series = Universitext | year = 2012 | publisher = Springer | location = New York, NY | isbn = 978-1-4614-0337-1 | doi = 10.1007/978-1-4614-0338-8 | zbl = 1239.15001

References

  1. Yasuda, M.. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl..
  2. {{harvnb. Fuhrmann. 2012
  3. Aoki, Masanao. (1983). "Notes on Economic Time Series Analysis : System Theoretic Perspectives". Springer.
  4. Aoki, Masanao. (1983). "Notes on Economic Time Series Analysis : System Theoretic Perspectives". Springer.
  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

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matrices-(mathematics)transforms