Companion matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial 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 square matrix 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 transpose of this matrix, C ( p ) T {\displaystyle C(p)^{T}} , which is more convenient for some purposes such as linear recurrence relations (see below).