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
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(R).
Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton theorem.
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
itself, the result is the zero matrix:
An polynomial annihilates
is also known as an annihilating polynomial.
Thus, the characteristic polynomial is a polynomial which annihilates
There is a unique monic polynomial of minimal degree which annihilates
Any polynomial which annihilates
(such as the characteristic polynomial) is a multiple of the minimal polynomial.
(the index of an eigenvalue is the size of its largest Jordan block).
[3] Matrix polynomials can be used to sum a matrix geometrical series as one would an ordinary geometric series, If
is nonsingular one can evaluate the expression for the sum