Matrix polynomial

Not to be confused with Polynomial matrix.

In mathematics, a matrix polynomial is a polynomial with matrices as variables. Examples include:

$P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},$

where P is a polynomial,

$P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},$

and I is the identity matrix.

$\left[A,B\right]=AB-BA,$

the commutator of A and B.

A matrix polynomial equation is an equation of the form P(A) = Q(A), which holds for the specific matrix A.
A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring M_n(R).