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Hankel matrix

<p>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a Hankel matrix (or <a href="/facts/Catalecticant/syyrJI25">catalecticant</a> matrix), named after <a href="/facts/Hermann_Hankel/rWmnbQC8">Hermann Hankel</a>, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example,
</p><p>
  
    
      
        
        
          
            [
            
              
                
                  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
                
              
            
            ]
          
        
        .
      
    
    {\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}}.}

</p><p>More generally, a Hankel matrix is any 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
 of the form
</p><p>
  
    
      
        A
        =
        
          
            [
            
              
                
                  
                    a
                    
                      0
                    
                  
                
                
                  
                    a
                    
                      1
                    
                  
                
                
                  
                    a
                    
                      2
                    
                  
                
                
                  …
                
                
                  
                    a
                    
                      n
                      −
                      1
                    
                  
                
              
              
                
                  
                    a
                    
                      1
                    
                  
                
                
                  
                    a
                    
                      2
                    
                  
                
                
                
                
                  ⋮
                
              
              
                
                  
                    a
                    
                      2
                    
                  
                
                
                
                
                
                  
                    a
                    
                      2
                      n
                      −
                      4
                    
                  
                
              
              
                
                  ⋮
                
                
                
                
                  
                    a
                    
                      2
                      n
                      −
                      4
                    
                  
                
                
                  
                    a
                    
                      2
                      n
                      −
                      3
                    
                  
                
              
              
                
                  
                    a
                    
                      n
                      −
                      1
                    
                  
                
                
                  …
                
                
                  
                    a
                    
                      2
                      n
                      −
                      4
                    
                  
                
                
                  
                    a
                    
                      2
                      n
                      −
                      3
                    
                  
                
                
                  
                    a
                    
                      2
                      n
                      −
                      2
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\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}}.}

</p><p>In terms of the components, if the 
  
    
      
        i
        ,
        j
      
    
    {\displaystyle i,j}
  
 element of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is denoted with 
  
    
      
        
          A
          
            i
            j
          
        
      
    
    {\displaystyle A_{ij}}
  
, and assuming 
  
    
      
        i
        ≤
        j
      
    
    {\displaystyle i\leq j}
  
, then we have 
  
    
      
        
          A
          
            i
            ,
            j
          
        
        =
        
          A
          
            i
            +
            k
            ,
            j
            −
            k
          
        
      
    
    {\displaystyle A_{i,j}=A_{i+k,j-k}}
  
 for all 
  
    
      
        k
        =
        0
        ,
        .
        .
        .
        ,
        j
        −
        i
        .
      
    
    {\displaystyle k=0,...,j-i.}

</p>