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,

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

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 $i\leq j$, 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 $H=TJ_{n}$ where $T$ is an $m\times n$ Toeplitz matrix.
  • If $T$ is real symmetric, then $H=TJ_{n}$ 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 $$f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},$$ 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 $${\begin{bmatrix}a_{1}&a_{2}&\ldots \\a_{2}&a_{3}&\ldots \\a_{3}&a_{4}&\ldots \\\vdots &\vdots &\ddots \end{bmatrix}}.$$ 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

Not to be confused with Hankel 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

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. 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.