Companion matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the <a href="/facts/Ferdinand_Georg_Frobenius/4cXXQyu5">Frobenius</a> companion matrix of the <a href="/facts/Monic_polynomial/4JzuIJ5R">monic polynomial</a>

p
        (
        x
        )
        =
        
          c
          
            0
          
        
        +
        
          c
          
            1
          
        
        x
        +
        ⋯
        +
        
          c
          
            n
            −
            1
          
        
        
          x
          
            n
            −
            1
          
        
        +
        
          x
          
            n
          
        
      
    
    {\displaystyle p(x)=c_{0}+c_{1}x+\cdots +c_{n-1}x^{n-1}+x^{n}}

is the <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> defined as

 
 
 
 C
 (
 p
 )
 =
 
 
 [
 
 
 
 0
 
 
 0
 
 
 …
 
 
 0
 
 
 −
 
 c
 
 0
 
 
 
 
 
 
 1
 
 
 0
 
 
 …
 
 
 0
 
 
 −
 
 c
 
 1
 
 
 
 
 
 
 0
 
 
 1
 
 
 …
 
 
 0
 
 
 −
 
 c
 
 2
 
 
 
 
 
 
 ⋮
 
 
 ⋮
 
 
 ⋱
 
 
 ⋮
 
 
 ⋮
 
 
 
 
 0
 
 
 0
 
 
 …
 
 
 1
 
 
 −
 
 c
 
 n
 −
 1
 
 
 
 
 
 ]
 
 
 .
 
 
 {\displaystyle C(p)={\begin{bmatrix}0&0&\dots &0&-c_{0}\\1&0&\dots &0&-c_{1}\\0&1&\dots &0&-c_{2}\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\dots &1&-c_{n-1}\end{bmatrix}}.}

Some authors use the <a href="/facts/Transpose/8wmsagGS">transpose</a> of this matrix, 
 
 
 
 C
 (
 p
 
 )
 
 T
 
 
 
 
 {\displaystyle C(p)^{T}}
 
, which is more convenient for some purposes such as linear <a href="/facts/Recurrence_relation/JolXC71M">recurrence relations</a> (see below).

 
 
 
 C
 (
 p
 )
 
 
 {\displaystyle C(p)}
 
 is defined from the coefficients of 
 
 
 
 p
 (
 x
 )
 
 
 {\displaystyle p(x)}
 
, while the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> as well as the <a href="/facts/Minimal_polynomial_(linear_algebra)/kv3lQb64">minimal polynomial</a> of 
 
 
 
 C
 (
 p
 )
 
 
 {\displaystyle C(p)}
 
 are equal to 
 
 
 
 p
 (
 x
 )
 
 
 {\displaystyle p(x)}
 
. In this sense, the matrix 
 
 
 
 C
 (
 p
 )
 
 
 {\displaystyle C(p)}
 
 and the polynomial 
 
 
 
 p
 (
 x
 )
 
 
 {\displaystyle p(x)}
 
 are "companions".

Companion matrix open-in-new

Companion matrix