Hilbert matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a Hilbert matrix, introduced by <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert</a> (1894), is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> with entries being the <a href="/facts/Unit_fraction/tjAfHzCH">unit fractions</a>

H
          
            i
            j
          
        
        =
        
          
            1
            
              i
              +
              j
              −
              1
            
          
        
        .
      
    
    {\displaystyle H_{ij}={\frac {1}{i+j-1}}.}

For example, this is the 5 × 5 Hilbert matrix:

H
        =
        
          
            [
            
              
                
                  1
                
                
                  
                    
                      1
                      2
                    
                  
                
                
                  
                    
                      1
                      3
                    
                  
                
                
                  
                    
                      1
                      4
                    
                  
                
                
                  
                    
                      1
                      5
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                  
                
                
                  
                    
                      1
                      3
                    
                  
                
                
                  
                    
                      1
                      4
                    
                  
                
                
                  
                    
                      1
                      5
                    
                  
                
                
                  
                    
                      1
                      6
                    
                  
                
              
              
                
                  
                    
                      1
                      3
                    
                  
                
                
                  
                    
                      1
                      4
                    
                  
                
                
                  
                    
                      1
                      5
                    
                  
                
                
                  
                    
                      1
                      6
                    
                  
                
                
                  
                    
                      1
                      7
                    
                  
                
              
              
                
                  
                    
                      1
                      4
                    
                  
                
                
                  
                    
                      1
                      5
                    
                  
                
                
                  
                    
                      1
                      6
                    
                  
                
                
                  
                    
                      1
                      7
                    
                  
                
                
                  
                    
                      1
                      8
                    
                  
                
              
              
                
                  
                    
                      1
                      5
                    
                  
                
                
                  
                    
                      1
                      6
                    
                  
                
                
                  
                    
                      1
                      7
                    
                  
                
                
                  
                    
                      1
                      8
                    
                  
                
                
                  
                    
                      1
                      9
                    
                  
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle H={\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}\\{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}\\{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}&{\frac {1}{9}}\end{bmatrix}}.}

The entries can also be defined by the integral

H
          
            i
            j
          
        
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          x
          
            i
            +
            j
            −
            2
          
        
        
        d
        x
        ,
      
    
    {\displaystyle H_{ij}=\int _{0}^{1}x^{i+j-2}\,dx,}

that is, as a <a href="/facts/Gramian_matrix/SGb2VGTU">Gramian matrix</a> for powers of x. It arises in the <a href="/facts/Least_squares/50jIwbxC">least squares</a> approximation of arbitrary functions by <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>.
The Hilbert matrices are canonical examples of <a href="/facts/Ill-conditioned/9Tjtk6wh">ill-conditioned</a> matrices, being notoriously difficult to use in <a href="/facts/Numerical_analysis/QT9cYa9z">numerical computation</a>. For example, the 2-norm <a href="/facts/Condition_number/9Tjtk6wh">condition number</a> of the matrix above is about 4.8×105.

Hilbert matrix open-in-new

Hilbert matrix