In mathematics, a matrix polynomial is a polynomial with matrices as variables. Given an ordinary, scalar-valued polynomial
this polynomial evaluated at a matrix A is
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 . 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 (such as the characteristic polynomial) is a multiple of the minimal polynomial.
It follows that given two polynomials P and Q, we have if and only if
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.
- Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) . Matrix Polynomials. Classics in Applied Mathematics. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 0-898716-81-0. Zbl 1170.15300.
- Higham, Nicholas J. (2000). Functions of Matrices: Theory and Computation. SIAM. ISBN 089-871-777-9..
- Horn, Roger A.; Johnson, Charles R. (1990). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6..