Matrix polynomial

In mathematics, a matrix polynomial is a polynomial with <a href="/facts/Square_matrix/5TP9mNNn">square matrices</a> as variables. Given an ordinary, scalar-valued polynomial

P
        (
        x
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          
            a
            
              i
            
          
          
            x
            
              i
            
          
        
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        x
        +
        
          a
          
            2
          
        
        
          x
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            n
          
        
        
          x
          
            n
          
        
        ,
      
    
    {\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}

this polynomial evaluated at a matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is

P
        (
        A
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          
            a
            
              i
            
          
          
            A
            
              i
            
          
        
        =
        
          a
          
            0
          
        
        I
        +
        
          a
          
            1
          
        
        A
        +
        
          a
          
            2
          
        
        
          A
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            n
          
        
        
          A
          
            n
          
        
        ,
      
    
    {\displaystyle P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},}

where 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>.
Note that 
 
 
 
 P
 (
 A
 )
 
 
 {\displaystyle P(A)}
 
 has the same dimension as 
 
 
 
 A
 
 
 {\displaystyle A}
 
.
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified <a href="/facts/Matrix_ring/oj5HBr8B">matrix ring</a> Mn(R).
Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of <a href="/facts/Linear_transformations/Q1PBKNVV">linear transformations</a> represented as matrices, most notably the <a href="/facts/Cayley%25E2%2580%2593Hamilton_theorem/pXdJXWkR">Cayley–Hamilton theorem</a>.

Matrix polynomial open-in-new

Matrix polynomial