Editing Hankel matrix

{{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 (mathematics)|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 Hankel matrix is [[symmetric matrix|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 number|real]] symmetric, then <math>H = T J_n</math> will have the same [[eigenvalue]]s 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]].

==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 C[[z^{-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 C[[z^{-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 C[[z^{-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 (linear algebra)|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 (statistics)|method of moments]] applied to polynomial distributions results in a Hankel matrix that needs to be [[inverse matrix|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 ==
*[[Richard P. Brent|Brent R.P.]] (1999), "Stability of fast algorithms for structured linear systems", ''Fast Reliable Algorithms for Matrices with Structure'' (editors&mdash;T. Kailath, A.H. Sayed), ch.4 ([[Society for Industrial and Applied Mathematics|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]]