Hankel matrix

<h2 id="properties">Properties</h2>
<ul><li>Any Hankel matrix is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a>.</li>
<li>Let 
  
    
      
        
          J
          
            n
          
        
      
    
    {\displaystyle J_{n}}
  
 be the 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Exchange_matrix/8zHvNqHP">exchange matrix</a>.  If 
  
    
      
        H
      
    
    {\displaystyle H}
  
 is an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
  
 Hankel matrix, then 
  
    
      
        H
        =
        T
        
          J
          
            n
          
        
      
    
    {\displaystyle H=TJ_{n}}
  
 where 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
  
 <a href="/facts/Toeplitz_matrix/WAjSdlaW">Toeplitz matrix</a>.
<ul><li>If 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is <a href="/facts/Real_number/R02gw5Pb">real</a> symmetric, then 
  
    
      
        H
        =
        T
        
          J
          
            n
          
        
      
    
    {\displaystyle H=TJ_{n}}
  
 will have the same <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> as 
  
    
      
        T
      
    
    {\displaystyle T}
  
 up to sign.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul></li>
<li>The <a href="/facts/Hilbert_matrix/60xPxyzt">Hilbert matrix</a> is an example of a Hankel matrix.</li>
<li>The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of a Hankel matrix is called a <a href="/facts/Catalecticant/syyrJI25">catalecticant</a>.</li></ul>
<h2 id="hankel-operator">Hankel operator</h2>
<p>Given a <a href="/facts/Formal_Laurent_series/CNmaG0Vk">formal Laurent series</a>

f
        (
        z
        )
        =
        
          ∑
          
            n
            =
            −
            ∞
          
          
            N
          
        
        
          a
          
            n
          
        
        
          z
          
            n
          
        
        ,
      
    
    {\displaystyle f(z)=\sum _{n=-\infty }^{N}a_{n}z^{n},}

the corresponding Hankel operator is defined as<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>

H
          
            f
          
        
        :
        
          C
        
        [
        z
        ]
        →
        
          
            z
          
          
            −
            1
          
        
        
          C
        
        [
        [
        
          z
          
            −
            1
          
        
        ]
        ]
        .
      
    
    {\displaystyle H_{f}:\mathbf {C} [z]\to \mathbf {z} ^{-1}\mathbf {C} [[z^{-1}]].}

This takes a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> 
  
    
      
        g
        ∈
        
          C
        
        [
        z
        ]
      
    
    {\displaystyle g\in \mathbf {C} [z]}
  
 and sends it to the product 
  
    
      
        f
        g
      
    
    {\displaystyle fg}
  
, but discards all powers of 
  
    
      
        z
      
    
    {\displaystyle z}
  
 with a non-negative exponent, so as to give an element in 
  
    
      
        
          z
          
            −
            1
          
        
        
          C
        
        [
        [
        
          z
          
            −
            1
          
        
        ]
        ]
      
    
    {\displaystyle z^{-1}\mathbf {C} [[z^{-1}]]}
  
, the <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> with strictly negative exponents. The map 
  
    
      
        
          H
          
            f
          
        
      
    
    {\displaystyle H_{f}}
  
 is in a natural way 
  
    
      
        
          C
        
        [
        z
        ]
      
    
    {\displaystyle \mathbf {C} [z]}
  
-linear, and its matrix with respect to the elements 
  
    
      
        1
        ,
        z
        ,
        
          z
          
            2
          
        
        ,
        ⋯
        ∈
        
          C
        
        [
        z
        ]
      
    
    {\displaystyle 1,z,z^{2},\dots \in \mathbf {C} [z]}
  
 and 
  
    
      
        
          z
          
            −
            1
          
        
        ,
        
          z
          
            −
            2
          
        
        ,
        ⋯
        ∈
        
          z
          
            −
            1
          
        
        
          C
        
        [
        [
        
          z
          
            −
            1
          
        
        ]
        ]
      
    
    {\displaystyle z^{-1},z^{-2},\dots \in z^{-1}\mathbf {C} [[z^{-1}]]}
  
 is the Hankel matrix

[
            
              
                
                  
                    a
                    
                      1
                    
                  
                
                
                  
                    a
                    
                      2
                    
                  
                
                
                  …
                
              
              
                
                  
                    a
                    
                      2
                    
                  
                
                
                  
                    a
                    
                      3
                    
                  
                
                
                  …
                
              
              
                
                  
                    a
                    
                      3
                    
                  
                
                
                  
                    a
                    
                      4
                    
                  
                
                
                  …
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle {\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 <a href="/facts/Theorem/f2Pe1FMD">theorem</a> due to <a href="/facts/Kronecker/1pYYP8jt">Kronecker</a> says that the <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a> of this matrix is finite precisely if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a>, that is, a fraction of two polynomials

f
        (
        z
        )
        =
        
          
            
              p
              (
              z
              )
            
            
              q
              (
              z
              )
            
          
        
        .
      
    
    {\displaystyle f(z)={\frac {p(z)}{q(z)}}.}

</p>
<h2 id="approximations">Approximations</h2>
<p>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 <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a> as a possible technique to approximate the action of the operator.
</p><p>Note that the matrix 
  
    
      
        A
      
    
    {\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 with AAK theory.
</p>
<h2 id="hankel-matrix-transform">Hankel matrix transform</h2>
<p class="note">Not to be confused with <a href="/facts/Hankel_transform/2UpX2jL8">Hankel transform</a>.</p>
<p>The Hankel matrix transform, or simply Hankel transform, of a <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
  
    
      
        
          b
          
            k
          
        
      
    
    {\displaystyle b_{k}}
  
 is the sequence of the determinants of the Hankel matrices formed from 
  
    
      
        
          b
          
            k
          
        
      
    
    {\displaystyle b_{k}}
  
. Given an integer 
  
    
      
        n
        >
        0
      
    
    {\displaystyle n>0}
  
, define the corresponding 
  
    
      
        (
        n
        ×
        n
        )
      
    
    {\displaystyle (n\times n)}
  
-dimensional Hankel matrix 
  
    
      
        
          B
          
            n
          
        
      
    
    {\displaystyle B_{n}}
  
 as having the matrix elements 
  
    
      
        [
        
          B
          
            n
          
        
        
          ]
          
            i
            ,
            j
          
        
        =
        
          b
          
            i
            +
            j
          
        
        .
      
    
    {\displaystyle [B_{n}]_{i,j}=b_{i+j}.}
  
 Then the sequence 
  
    
      
        
          h
          
            n
          
        
      
    
    {\displaystyle h_{n}}
  
 given by

h
          
            n
          
        
        =
        det
        
          B
          
            n
          
        
      
    
    {\displaystyle h_{n}=\det B_{n}}

is the Hankel transform of the sequence 
  
    
      
        
          b
          
            k
          
        
        .
      
    
    {\displaystyle b_{k}.}
  
 The Hankel transform is invariant under the <a href="/facts/Binomial_transform/Wu345jgq">binomial transform</a> of a sequence. That is, if one writes

c
          
            n
          
        
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          
            
              (
            
            
              n
              k
            
            
              )
            
          
        
        
          b
          
            k
          
        
      
    
    {\displaystyle c_{n}=\sum _{k=0}^{n}{n \choose k}b_{k}}

as the binomial transform of the sequence 
  
    
      
        
          b
          
            n
          
        
      
    
    {\displaystyle b_{n}}
  
, then one has 
  
    
      
        det
        
          B
          
            n
          
        
        =
        det
        
          C
          
            n
          
        
        .
      
    
    {\displaystyle \det B_{n}=\det C_{n}.}

</p>
<h2 id="applications-of-hankel-matrices">Applications of Hankel matrices</h2>
<p>Hankel matrices are formed when, given a sequence of output data, a realization of an underlying state-space or <a href="/facts/Hidden_Markov_model/ur1zTAhP">hidden Markov model</a> is desired.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  The singular value decomposition of the Hankel matrix provides a means of computing the <i>A</i>, <i>B</i>, and <i>C</i> matrices which define the state-space realization.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The Hankel matrix formed from the signal has been found useful for decomposition of non-stationary signals and time-frequency representation.
</p>
<h3>Method of moments for polynomial distributions</h3>
<p>The <a href="/facts/Method_of_moments_(statistics)/Qfso3fMe">method of moments</a> applied to polynomial distributions results in a Hankel matrix that needs to be <a href="/facts/Inverse_matrix/fPqXk3V8">inverted</a> in order to obtain the weight parameters of the polynomial distribution approximation.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Positive Hankel matrices and the Hamburger moment problems</h3>
<p class="note">Further information: <a href="/facts/Hamburger_moment_problem/0cLzE4tD">Hamburger moment problem</a></p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cauchy_matrix/bMQ68GnS">Cauchy matrix</a></li>
<li><a href="/facts/Jacobi_operator/FtG47pD7">Jacobi operator</a></li>
<li><a href="/facts/Toeplitz_matrix/WAjSdlaW">Toeplitz matrix</a>, an "upside down" (that is, row-reversed) Hankel matrix</li>
<li><a href="/facts/Vandermonde_matrix/8oSbUk6u">Vandermonde matrix</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Richard_P._Brent/JYLmuA0M">Brent R.P.</a> (1999), "Stability of fast algorithms for structured linear systems", <i>Fast Reliable Algorithms for Matrices with Structure</i> (editors—T. Kailath, A.H. Sayed), ch.4 (<a href="/facts/Society_for_Industrial_and_Applied_Mathematics/2HY4T5WY">SIAM</a>).</li>
<li>Fuhrmann, Paul A. (2012). <i>A polynomial approach to linear algebra</i>. Universitext (2 ed.). New York, NY: Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4614-0338-8">10.1007/978-1-4614-0338-8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4614-0337-1. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1239.15001">1239.15001</a>.</li></ul>
<ul><li><a href="/facts/Victor_Pan/Eb9w9CIu">Victor Y. Pan</a> (2001). <i>Structured matrices and polynomials: unified superfast algorithms</i>. <a href="/facts/Birkh%25C3%25A4user/brvgnrnJ">Birkhäuser</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0817642404.</li>
<li><a href="/facts/Jonathan_Partington/Aa3fB6ej">J.R. Partington</a> (1988). <i>An introduction to Hankel operators</i>. LMS Student Texts. Vol. 13. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-36791-3.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Yasuda, M. (2003). "A Spectral Characterization of Hermitian Centrosymmetric and Hermitian Skew-Centrosymmetric K-Matrices". SIAM J. Matrix Anal. Appl. 25 (3): 601–605. doi:10.1137/S0895479802418835. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Fuhrmann 2012, §8.3 - 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. <a href="https://doi.org/10.1007%2F978-1-4614-0338-8" target="_blank">https://doi.org/10.1007%2F978-1-4614-0338-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Aoki, Masanao (1983). "Prediction of Time Series". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 38–47. ISBN 0-387-12696-1. <a href="0-387-12696-1" target="_blank">0-387-12696-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Aoki, Masanao (1983). "Rank determination of Hankel matrices". Notes on Economic Time Series Analysis : System Theoretic Perspectives. New York: Springer. pp. 67–68. ISBN 0-387-12696-1. <a href="0-387-12696-1" target="_blank">0-387-12696-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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 <a href="https://doi.org/10.1371/journal.pone.0174573" target="_blank">https://doi.org/10.1371/journal.pone.0174573</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Hankel matrix open-in-new

Hankel matrix