In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. 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 ring R, the rings M n ( R [ X ] ) {\displaystyle M_{n}(R[X])} and ( M n ( R ) ) [ X ] {\displaystyle (M_{n}(R))[X]} are isomorphic.