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, $$\left[\begin{array}{ccccc}a& b& c& d& e\\ b& c& d& e& f\\ c& d& e& f& g\\ d& e& f& g& h\\ e& f& g& h& i\\ \end{array}\right].$$

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

$$A=\left[\begin{array}{ccccc}{a}_0& a_1& a_2& \dots & a_{n-1}\\ a_1& a_2& & & ⋮\\ a_2& & & & a_{2n-4}\\ ⋮& & & a_{2n-4}& a_{2n-3}\\ a_{n-1}& \dots & a_{2n-4}& a_{2n-3}& a_{2n-2}\end{array}\right].$$

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

• Any 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.
• 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

$f(z)=\sum _{n=-\mathrm{\infty }}^N{a}_n{z}^n$

the corresponding Hankel operator is defined as

$H_f:\mathbf{𝐂}[z]\to {\mathbf{𝐳}}^{-1}\mathbf{𝐂}{z}^{-1},$

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 $z^{-1}\mathbf{𝐂}{z}^{-1}$, 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 $z^{-1},z^{-2},\dots \in z^{-1}\mathbf{𝐂}{z}^{-1}$ is the Hankel matrix

$$\left[\begin{array}{ccc}{a}_1& a_2& \dots \\ a_2& a_3& \dots \\ a_3& a_4& \dots \\ ⋮& ⋮& \ddots \end{array}\right].$$

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

$c_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})b_k$

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. 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

Further information: Hamburger moment problem

See also

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

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. 

• 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.