Inlinear algebra, aHankel matrix (orcatalecticant matrix), named afterHermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example,

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

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

• Any Hankel matrix issymmetric.
• Let${\displaystyle J_n}$ be the${\displaystyle n\times n}$exchange matrix. If${\displaystyle H}$ is an${\displaystyle m\times n}$ Hankel matrix, then${\displaystyle H=T{J}_n}$ where${\displaystyle T}$ is an${\displaystyle m\times n}$Toeplitz matrix.
  • If${\displaystyle T}$ isreal symmetric, then${\displaystyle H=T{J}_n}$ will have the sameeigenvalues as${\displaystyle T}$ up to sign.^[1]
• TheHilbert matrix is an example of a Hankel matrix.
• Thedeterminant of a Hankel matrix is called acatalecticant.

Given aformal Laurent series$${\displaystyle f(z)=\sum _{n=-\mathrm{\infty }}^N{a}_n{z}^n,}$$the correspondingHankel operator is defined as^[2]$${\displaystyle H_f:\mathbf{C}[z]\to {\mathbf{z}}^{-1}\mathbf{C}[[z^{-1}]].}$$This takes apolynomial${\displaystyle g\in \mathbf{C}[z]}$ 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${\displaystyle z^{-1}\mathbf{C}[[z^{-1}]]}$, theformal 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${\displaystyle 1,z,z^2,\cdots \in \mathbf{C}[z]}$ and${\displaystyle z^{-1},z^{-2},\cdots \in z^{-1}\mathbf{C}[[z^{-1}]]}$ is the Hankel matrixAny Hankel matrix arises in this way. Atheorem due toKronecker says that therank of this matrix is finite precisely if${\displaystyle f}$ is arational function, that is, a fraction of two polynomials$${\displaystyle f(z)=\frac{p(z)}{q(z)}.}$$

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 suggestssingular 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 withAAK theory.

TheHankel matrix transform, or simplyHankel transform, of asequence${\displaystyle b_k}$ 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 thebinomial transform of a sequence. That is, if one writes$${\displaystyle c_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})b_k}$$as the binomial transform of the sequence${\displaystyle b_n}$, then one has${\displaystyle det{B}_n=det{C}_n.}$

Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space orhidden Markov model is desired.^[3] The singular value decomposition of the Hankel matrix provides a means of computing theA,B, andC 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.

Themethod of moments applied to polynomial distributions results in a Hankel matrix that needs to beinverted in order to obtain the weight parameters of the polynomial distribution approximation.^[5]

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

• 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.

• Matrices
• Transforms

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