Hankel matrix

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,

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

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

$${\displaystyle A={\begin{bmatrix}a_{0}&a_{1}&a_{2}&\ldots &a_{n-1}\\a_{1}&a_{2}&&&\vdots \\a_{2}&&&&a_{2n-4}\\\vdots &&&a_{2n-4}&a_{2n-3}\\a_{n-1}&\ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}\end{bmatrix}}.}$$

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

Properties

• Any Hankel matrix is symmetric.
• Let ${\displaystyle J_{n}}$  be the ${\displaystyle n\times n}$  exchange matrix. If ${\displaystyle H}$  is an ${\displaystyle m\times n}$  Hankel matrix, then ${\displaystyle H=TJ_{n}}$  where ${\displaystyle T}$  is an ${\displaystyle m\times n}$  Toeplitz matrix.
  • If ${\displaystyle T}$  is real symmetric, then ${\displaystyle H=TJ_{n}}$  will have the same eigenvalues as ${\displaystyle T}$  up to sign.^[1]
• The Hilbert matrix is an example of a Hankel matrix.
• The determinant of a Hankel matrix is called a catalecticant.

Hankel operator

Given a formal Laurent series

${\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n}}$ 

the corresponding Hankel operator is defined as^[2]

${\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]],}$ 

This takes a polynomial ${\displaystyle g\in \mathbf {C} [z]}$  and sends it to the product ${\displaystyle fg}$ , but discards all powers of ${\displaystyle z}$  with a non-negative exponent, so as to give an element in ${\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}$ , the formal power series with strictly negative exponents. The map ${\displaystyle H_{f}}$  is in a natural way ${\displaystyle \mathbf {C} [z]}$ -linear, and its matrix with respect to the elements ${\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}$  and ${\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}$  is the Hankel matrix

$${\displaystyle {\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.}$$

 

Any Hankel matrix arises in this way. A theorem due to Kronecker says that the rank of this matrix is finite precisely if ${\displaystyle f}$  is a rational function; that is, a fraction of two polynomials

${\displaystyle f(z)={\frac {p(z)}{q(z)}}.}$ 

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 ${\displaystyle A}$  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

Not to be confused with Hankel transform.

The Hankel matrix transform, or simply Hankel transform, of a sequence ${\displaystyle b_{k}}$  is the sequence of the determinants of the Hankel matrices formed from ${\displaystyle b_{k}}$ . Given an integer ${\displaystyle n>0}$ , define the corresponding ${\displaystyle n\times n}$ –dimensional Hankel matrix ${\displaystyle B_{n}}$  as having the matrix elements ${\displaystyle [B_{n}]_{i,j}=b_{i+j}.}$  Then, the sequence ${\displaystyle h_{n}}$  given by

${\displaystyle h_{n}=\det B_{n}}$ 

is the Hankel transform of the sequence ${\displaystyle b_{k}.}$  The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes

${\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}$ 

as the binomial transform of the sequence ${\displaystyle b_{n}}$ , then one has ${\displaystyle \det B_{n}=\det C_{n}.}$ 

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.^[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

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

Further information: Hamburger moment problem

See also

• Cauchy matrix
• Jacobi operator
• Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix
• Vandermonde matrix

Notes

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. Fuhrmann 2012, §8.3
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.
4. 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.
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

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.