Matrix polynomial

Not to be confused with Polynomial matrix.

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

$P(A) = \sum_{i=0}^n{ a_i A^i} =a_0 I %2B a_1 A %2B a_2 A^2 %2B \cdots %2B a_n A^n,$

where P is a polynomial,

$P(x) = \sum_{i=0}^n{ a_i x^i} =a_0 %2B a_1 x%2B a_2 x^2 %2B \cdots %2B a_n x^n,$

and I is the identity matrix.

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

the commutator of A and B.

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. If $P(A) = Q(A)$ , (where A is a matrix over a field), then the eigenvalues of A satisfy the characteristic equation $P(\lambda) = Q(\lambda)$ .
A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring M_n(R).