Cauchy matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Cauchy matrix, named after <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a>, is an m×n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> with elements aij in the form

a
          
            i
            j
          
        
        =
        
          
            1
            
              
                x
                
                  i
                
              
              −
              
                y
                
                  j
                
              
            
          
        
        ;
        
        
          x
          
            i
          
        
        −
        
          y
          
            j
          
        
        ≠
        0
        ,
        
        1
        ≤
        i
        ≤
        m
        ,
        
        1
        ≤
        j
        ≤
        n
      
    
    {\displaystyle a_{ij}={\frac {1}{x_{i}-y_{j}}};\quad x_{i}-y_{j}\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n}

where 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
 and 
 
 
 
 
 y
 
 j
 
 
 
 
 {\displaystyle y_{j}}
 
 are elements of a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
, and 
 
 
 
 (
 
 x
 
 i
 
 
 )
 
 
 {\displaystyle (x_{i})}
 
 and 
 
 
 
 (
 
 y
 
 j
 
 
 )
 
 
 {\displaystyle (y_{j})}
 
 are <a href="/facts/Injective/5ES6tqf5">injective</a> sequences (they contain distinct elements).

Cauchy matrix open-in-new

Cauchy matrix