Polynomial matrix

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:

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

where ${\displaystyle A(i)}$ denotes a matrix of constant coefficients, and ${\displaystyle A(p)}$ is non-zero. An example 3×3 polynomial matrix, degree 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 ${\displaystyle M_{n}(R[X])}$ and ${\displaystyle (M_{n}(R))[X]}$ are isomorphic.