Hankel matrix

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,

$[\begin{matrix} a & b & c & d & e \\ b & c & d & e & f \\ c & d & e & f & g \\ d & e & f & g & h \\ e & f & g & h & i \end{matrix}] .$

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

$A = [\begin{matrix} a_{0} & a_{1} & a_{2} & \dots & a_{n - 1} \\ a_{1} & a_{2} & ⋮ \\ a_{2} & a_{2 n - 4} \\ ⋮ & a_{2 n - 4} & a_{2 n - 3} \\ a_{n - 1} & \dots & a_{2 n - 4} & a_{2 n - 3} & a_{2 n - 2} \end{matrix}] .$

In terms of the components, if the $i, j$ element of $A$ is denoted with $A_{i j}$ , and assuming $i \leq j$ , then we have $A_{i, j} = A_{i + k, j - k}$ for all $k = 0, ..., j - i .$

Properties

Any square Hankel matrix is symmetric.
Let $J_{n}$ be the $n \times n$ exchange matrix. If $H$ is an $m \times n$ Hankel matrix, then $H = T J_{n}$ where $T$ is an $m \times n$ Toeplitz matrix.
If $T$ is real symmetric, then $H = T J_{n}$ will have the same eigenvalues as $T$ up to sign.^[1]
The Hilbert matrix is an example of a Hankel matrix.
The determinant of a Hankel matrix is called a catalecticant.
If $H$ is an $n \times n$ Hankel matrix, then $H = V^{T} D V$ where $V$ is a confluent Vandermonde matrix and $D$ is a block diagonal matrix, with symmetric and upper anti-triangular blocks^[2].

Hankel operator

Given a formal Laurent series $f (z) = \sum_{n = - \infty}^{N} a_{n} z^{n},$ the corresponding Hankel operator is defined as^[3] $H_{f} : 𝐂 [z] \to z^{- 1} 𝐂 [[z^{- 1}]] .$ This takes a polynomial $g \in 𝐂 [z]$ and sends it to the product $f g$ , but discards all powers of $z$ with a non-negative exponent, so as to give an element in $z^{- 1} 𝐂 [[z^{- 1}]]$ , the formal power series with strictly negative exponents. The map $H_{f}$ is in a natural way $𝐂 [z]$ -linear, and its matrix with respect to the elements $1, z, z^{2}, \dots \in 𝐂 [z]$ and $z^{- 1}, z^{- 2}, \dots \in z^{- 1} 𝐂 [[z^{- 1}]]$ is the Hankel matrix $[\begin{matrix} a_{- 1} & a_{- 2} & \dots \\ a_{- 2} & a_{- 3} & \dots \\ a_{- 3} & a_{- 4} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}] .$ 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 $c_{n} = \sum_{k = 0}^{n} (\binom{n}{k}) b_{k}$ 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.^[4] 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.^[5] 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.^[6]

Positive Hankel matrices and the Hamburger moment problems

Notes

↑ 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.
↑ Boley, D.L.; F.T., Luk; D., Vandevoorde (1997). "Vandermonde factorization of a Hankel matrix". Proceedings of the Workshop on Scientific Computing : Hong Kong, 10-12 March. pp. 27–39. ISBN 978-981-3083-60-8.
↑ 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

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.

[simax1-1] 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] Boley, D.L.; F.T., Luk; D., Vandevoorde (1997). "Vandermonde factorization of a Hankel matrix". Proceedings of the Workshop on Scientific Computing : Hong Kong, 10-12 March. pp. 27–39. ISBN 978-981-3083-60-8.

[3] Fuhrmann 2012, §8.3

[4] 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.

[5] 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.

[PolyD2-6] 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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