Polynomial matrix

In mathematics, a polynomial matrix is a matrix whose elements are polynomials of a single variable.

A polynomial matrix A of degree p is defined as:

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

where $A(i)$ denotes a matrix of constant coefficients, and $A(p)$ is non-zero. Thus a polynomial matrix is the matrix-equivalent of a polynomial, with each element of the matrix satisfying the definition of a polynomial of degree p.

An example 3×3 polynomial matrix, degree 2:

$P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}$

Note that this is not to be confused with a monomial matrix, which is simply a matrix with exactly one non-zero entry in each row and column.