Polynomial matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a polynomial matrix or matrix of polynomials is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> whose elements are univariate or multivariate <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.
A univariate polynomial matrix P of degree p is defined as:

P
        =
        
          ∑
          
            n
            =
            0
          
          
            p
          
        
        A
        (
        n
        )
        
          x
          
            n
          
        
        =
        A
        (
        0
        )
        +
        A
        (
        1
        )
        x
        +
        A
        (
        2
        )
        
          x
          
            2
          
        
        +
        ⋯
        +
        A
        (
        p
        )
        
          x
          
            p
          
        
      
    
    {\displaystyle P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p}}

where 
 
 
 
 A
 (
 i
 )
 
 
 {\displaystyle A(i)}
 
 denotes a matrix of constant coefficients, and 
 
 
 
 A
 (
 p
 )
 
 
 {\displaystyle A(p)}
 
 is non-zero. 
An example 3×3 polynomial matrix, degree 2:

P
        =
        
          
            (
            
              
                
                  1
                
                
                  
                    x
                    
                      2
                    
                  
                
                
                  x
                
              
              
                
                  0
                
                
                  2
                  x
                
                
                  2
                
              
              
                
                  3
                  x
                  +
                  2
                
                
                  
                    x
                    
                      2
                    
                  
                  −
                  1
                
                
                  0
                
              
            
            )
          
        
        =
        
          
            (
            
              
                
                  1
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  2
                
              
              
                
                  2
                
                
                  −
                  1
                
                
                  0
                
              
            
            )
          
        
        +
        
          
            (
            
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  2
                
                
                  0
                
              
              
                
                  3
                
                
                  0
                
                
                  0
                
              
            
            )
          
        
        x
        +
        
          
            (
            
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
            
            )
          
        
        
          x
          
            2
          
        
        .
      
    
    {\displaystyle P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.}

We can express this by saying that for a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R, the rings 
 
 
 
 
 M
 
 n
 
 
 (
 R
 [
 X
 ]
 )
 
 
 {\displaystyle M_{n}(R[X])}
 
 and

(
 
 M
 
 n
 
 
 (
 R
 )
 )
 [
 X
 ]
 
 
 {\displaystyle (M_{n}(R))[X]}
 
 are <a href="/facts/Ring_homomorphism/UptcMpFk">isomorphic</a>.

Polynomial matrix open-in-new

Polynomial matrix