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, [math]\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}. }[/math]

More generally, a Hankel matrix is any [math]\displaystyle{ n \times n }[/math] matrix [math]\displaystyle{ A }[/math] of the form

[math]\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}. }[/math]

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

Properties

• Any Hankel matrix is symmetric.
• Let [math]\displaystyle{ J_n }[/math] be the [math]\displaystyle{ n \times n }[/math] exchange matrix. If [math]\displaystyle{ H }[/math] is an [math]\displaystyle{ m \times n }[/math] Hankel matrix, then [math]\displaystyle{ H = T J_n }[/math] where [math]\displaystyle{ T }[/math] is an [math]\displaystyle{ m \times n }[/math] Toeplitz matrix.
  • If [math]\displaystyle{ T }[/math] is real symmetric, then [math]\displaystyle{ H = T J_n }[/math] will have the same eigenvalues as [math]\displaystyle{ T }[/math] 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

[math]\displaystyle{ f(z) = \sum_{n=-\infty}^N a_n z^n }[/math]

the corresponding Hankel operator is defined as

[math]\displaystyle{ H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf Cz^{-1}, }[/math]

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

[math]\displaystyle{ \begin{bmatrix} a_1 & a_2 & \ldots \\ a_2 & a_3 & \ldots \\ a_3 & a_4 & \ldots \\ \vdots & \vdots & \ddots \end{bmatrix}. }[/math]

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

[math]\displaystyle{ f(z) = \frac{p(z)}{q(z)}. }[/math]

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 [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ b_k }[/math] is the sequence of the determinants of the Hankel matrices formed from [math]\displaystyle{ b_k }[/math]. Given an integer [math]\displaystyle{ n\gt 0 }[/math], define the corresponding [math]\displaystyle{ n\times n }[/math]–dimensional Hankel matrix [math]\displaystyle{ B_n }[/math] as having the matrix elements [math]\displaystyle{ [B_n]_{i,j}=b_{i+j}. }[/math] Then, the sequence [math]\displaystyle{ h_n }[/math] given by

[math]\displaystyle{ h_n = \det B_n }[/math]

is the Hankel transform of the sequence [math]\displaystyle{ b_k. }[/math] The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes

[math]\displaystyle{ c_n = \sum_{k=0}^n {n \choose k} b_k }[/math]

as the binomial transform of the sequence [math]\displaystyle{ b_n }[/math], then one has [math]\displaystyle{ \det B_n = \det C_n. }[/math]

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.