Matrix polynomial


In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial
this polynomial evaluated at a matrix A is
where I is the identity matrix.
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn.

Characteristic and minimal polynomial

The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by. The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A itself, the result is the zero matrix:. The characteristic polynomial is thus a polynomial which annihilates A.
There is a unique monic polynomial of minimal degree which annihilates A; this polynomial is the minimal polynomial. Any polynomial which annihilates A is a multiple of the minimal polynomial.
It follows that given two polynomials P and Q, we have if and only if
where denotes the jth derivative of P and are the eigenvalues of A with corresponding indices .

Matrix geometrical series

Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series,
If I − A is nonsingular one can evaluate the expression for the sum S.