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"("R").
Characteristic and minimal polynomial.
The characteristic polynomial of a matrix "A" is a scalar-valued polynomial, defined by formula_3. 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: formula_4. 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" (such as the characteristic polynomial) is a multiple of the minimal polynomial.
It follows that given two polynomials "P" and "Q", we have formula_5 if and only if 
where formula_7 denotes the "j"th derivative of "P" and formula_8 are the eigenvalues of "A" with corresponding indices formula_9 (the index of an eigenvalue is the size of its largest Jordan block).
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".