Alternant matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, an alternant matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of a square alternant matrix. 
Generally, if 
 
 
 
 
 f
 
 1
 
 
 ,
 
 f
 
 2
 
 
 ,
 …
 ,
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{1},f_{2},\dots ,f_{n}}
 
 are functions from a set 
 
 
 
 X
 
 
 {\displaystyle X}
 
 to a field 
 
 
 
 F
 
 
 {\displaystyle F}
 
, and 
 
 
 
 
 
 α
 
 1
 
 
 ,
 
 α
 
 2
 
 
 ,
 …
 ,
 
 α
 
 m
 
 
 
 ∈
 X
 
 
 {\displaystyle {\alpha _{1},\alpha _{2},\ldots ,\alpha _{m}}\in X}
 
, then the alternant matrix has size 
 
 
 
 m
 ×
 n
 
 
 {\displaystyle m\times n}
 
 and is defined by

M
        =
        
          
            [
            
              
                
                  
                    f
                    
                      1
                    
                  
                  (
                  
                    α
                    
                      1
                    
                  
                  )
                
                
                  
                    f
                    
                      2
                    
                  
                  (
                  
                    α
                    
                      1
                    
                  
                  )
                
                
                  ⋯
                
                
                  
                    f
                    
                      n
                    
                  
                  (
                  
                    α
                    
                      1
                    
                  
                  )
                
              
              
                
                  
                    f
                    
                      1
                    
                  
                  (
                  
                    α
                    
                      2
                    
                  
                  )
                
                
                  
                    f
                    
                      2
                    
                  
                  (
                  
                    α
                    
                      2
                    
                  
                  )
                
                
                  ⋯
                
                
                  
                    f
                    
                      n
                    
                  
                  (
                  
                    α
                    
                      2
                    
                  
                  )
                
              
              
                
                  
                    f
                    
                      1
                    
                  
                  (
                  
                    α
                    
                      3
                    
                  
                  )
                
                
                  
                    f
                    
                      2
                    
                  
                  (
                  
                    α
                    
                      3
                    
                  
                  )
                
                
                  ⋯
                
                
                  
                    f
                    
                      n
                    
                  
                  (
                  
                    α
                    
                      3
                    
                  
                  )
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    f
                    
                      1
                    
                  
                  (
                  
                    α
                    
                      m
                    
                  
                  )
                
                
                  
                    f
                    
                      2
                    
                  
                  (
                  
                    α
                    
                      m
                    
                  
                  )
                
                
                  ⋯
                
                
                  
                    f
                    
                      n
                    
                  
                  (
                  
                    α
                    
                      m
                    
                  
                  )
                
              
            
            ]
          
        
      
    
    {\displaystyle M={\begin{bmatrix}f_{1}(\alpha _{1})&f_{2}(\alpha _{1})&\cdots &f_{n}(\alpha _{1})\\f_{1}(\alpha _{2})&f_{2}(\alpha _{2})&\cdots &f_{n}(\alpha _{2})\\f_{1}(\alpha _{3})&f_{2}(\alpha _{3})&\cdots &f_{n}(\alpha _{3})\\\vdots &\vdots &\ddots &\vdots \\f_{1}(\alpha _{m})&f_{2}(\alpha _{m})&\cdots &f_{n}(\alpha _{m})\\\end{bmatrix}}}

or, more compactly, 
 
 
 
 
 M
 
 i
 j
 
 
 =
 
 f
 
 j
 
 
 (
 
 α
 
 i
 
 
 )
 
 
 {\displaystyle M_{ij}=f_{j}(\alpha _{i})}
 
. (Some authors use the <a href="/facts/Transpose/8wmsagGS">transpose</a> of the above matrix.) Examples of alternant matrices include <a href="/facts/Vandermonde_matrix/8oSbUk6u">Vandermonde matrices</a>, for which 
 
 
 
 
 f
 
 j
 
 
 (
 α
 )
 =
 
 α
 
 j
 −
 1
 
 
 
 
 {\displaystyle f_{j}(\alpha )=\alpha ^{j-1}}
 
, and <a href="/facts/Moore_matrices/UXb29gHE">Moore matrices</a>, for which 
 
 
 
 
 f
 
 j
 
 
 (
 α
 )
 =
 
 α
 
 
 q
 
 j
 −
 1
 
 
 
 
 
 
 {\displaystyle f_{j}(\alpha )=\alpha ^{q^{j-1}}}
 
.

Alternant matrix open-in-new

Alternant matrix