{{Short description|Square matrix in which each ascending skew-diagonal from left to right is constant}} In linear algebra, a '''Hankel matrix''' (or '''catalecticant matrix'''), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example,
<math display=block>\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>n \times n</math> matrix <math>A</math> of the form
<math display=block>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>i,j</math> element of <math>A</math> is denoted with <math>A_{ij}</math>, and assuming <math>i \le j</math>, then we have <math>A_{i,j} = A_{i+k,j-k}</math> for all <math>k = 0,...,j-i.</math>
==Properties== * Any square Hankel matrix is symmetric. * Let <math>J_n</math> be the <math>n \times n</math> exchange matrix. If <math>H</math> is an <math>m \times n</math> Hankel matrix, then <math>H = T J_n</math> where <math>T</math> is an <math>m \times n</math> Toeplitz matrix. * If <math>T</math> is real symmetric, then <math>H = T J_n</math> will have the same eigenvalues as <math>T</math> up to sign.<ref name="simax1">{{cite journal | last = Yasuda | first = M. | title = A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices | journal = SIAM J. Matrix Anal. Appl. | volume = 25 | issue = 3 | pages = 601–605 | year = 2003 | doi = 10.1137/S0895479802418835}}</ref> * The Hilbert matrix is an example of a Hankel matrix. * The determinant of a Hankel matrix is called a catalecticant. * If <math>H</math> is an <math>n \times n</math> Hankel matrix, then <math>H=V^TDV</math> where <math>V</math> is a confluent Vandermonde matrix and <math>D</math> is a block diagonal matrix, with symmetric and upper anti-triangular blocks<ref>{{cite conference|last1=Boley|first1=D.L.|last2=F.T.|first2=Luk|last3=D.|first3=Vandevoorde|contribution=Vandermonde factorization of a Hankel matrix|pages=27-39|title=Proceedings of the Workshop on Scientific Computing : Hong Kong, 10-12 March|year=1997|isbn=978-981-3083-60-8|url=https://link.springer.com/book/9789813083608}}</ref>.
==Hankel operator== Given a formal Laurent series <math display="block"> f(z) = \sum_{n=-\infty}^N a_n z^n, </math> the corresponding '''Hankel operator''' is defined as<ref>{{harvnb|Fuhrmann|2012|loc=§8.3}}</ref> <math display="block"> H_f : \mathbf C[z] \to \mathbf z^{-1} \mathbf Cz^{-1}. </math> This takes a polynomial <math>g \in \mathbf C[z]</math> and sends it to the product <math>fg</math>, but discards all powers of <math>z</math> with a non-negative exponent, so as to give an element in <math>z^{-1} \mathbf Cz^{-1}</math>, the formal power series with strictly negative exponents. The map <math>H_f</math> is in a natural way <math>\mathbf C[z]</math>-linear, and its matrix with respect to the elements <math>1, z, z^2, \dots \in \mathbf C[z]</math> and <math>z^{-1}, z^{-2}, \dots \in z^{-1}\mathbf Cz^{-1}</math> is the Hankel matrix <math display=block>\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>f</math> is a rational function, that is, a fraction of two polynomials <math display="block"> 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>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== {{Distinguish|Hankel transform}}
The '''Hankel matrix transform''', or simply '''Hankel transform''', of a sequence <math>b_k</math> is the sequence of the determinants of the Hankel matrices formed from <math>b_k</math>. Given an integer <math>n > 0</math>, define the corresponding <math>(n \times n)</math>-dimensional Hankel matrix <math>B_n</math> as having the matrix elements <math>[B_n]_{i,j} = b_{i+j}.</math> Then the sequence <math>h_n</math> given by <math display="block"> h_n = \det B_n </math> is the Hankel transform of the sequence <math>b_k.</math> The Hankel transform is invariant under the binomial transform of a sequence. That is, if one writes <math display="block"> c_n = \sum_{k=0}^n {n \choose k} b_k </math> as the binomial transform of the sequence <math>b_n</math>, then one has <math>\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.<ref>{{cite book |first=Masanao |last=Aoki |author-link=Masanao Aoki |chapter=Prediction of Time Series |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=38–47 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA38 }}</ref> 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.<ref>{{cite book |first=Masanao |last=Aoki |chapter=Rank determination of Hankel matrices |title=Notes on Economic Time Series Analysis : System Theoretic Perspectives |location=New York |publisher=Springer |year=1983 |isbn=0-387-12696-1 |pages=67–68 |chapter-url=https://books.google.com/books?id=l_LsCAAAQBAJ&pg=PA67 }}</ref> 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.<ref name="PolyD2">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</ref>
=== Positive Hankel matrices and the Hamburger moment problems === {{Further|Hamburger moment problem}}
==See also== * Cauchy matrix * Jacobi operator * Toeplitz matrix, an "upside down" (that is, row-reversed) Hankel matrix * Vandermonde matrix
== Notes == {{Reflist}}
== 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). *{{cite book | last = Fuhrmann | first = Paul A. | title = A polynomial approach to linear algebra | edition = 2 | series = Universitext | year = 2012 | publisher = Springer | location = New York, NY | isbn = 978-1-4614-0337-1 | doi = 10.1007/978-1-4614-0338-8 | zbl = 1239.15001 }}
* {{cite book | title=Structured matrices and polynomials: unified superfast algorithms | author=Victor Y. Pan | author-link=Victor Pan | publisher=Birkhäuser | year=2001 | isbn=0817642404 }} * {{cite book | title=An introduction to Hankel operators | author=J.R. Partington | author-link=Jonathan Partington | series=LMS Student Texts | volume=13 | publisher=Cambridge University Press | year=1988 | isbn=0-521-36791-3 }}
{{Matrix classes}} {{Authority control}}
Category:Matrices (mathematics) Category:Transforms