# Matrix polynomial

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.
$\left[A,B\right] = A B - B A,$
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[disputed ] $P(\lambda) = Q(\lambda)$.
A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring Mn(R).

## Matrix geometrical series

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,

$S=I+A+A^2+\cdots +A^n$
$AS=A+A^2+A^3+\cdots +A^{n+1}$
$(I-A)S=S-AS=I-A^{n+1}$
$S=(I-A)^{-1}(I-A^{n+1})$

If I-A is nonsingular one can evaluate the expression for the sum S.