In mathematics, a matrix polynomial is a polynomial with matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 as variables. Examples include:

where P is a polynomial,

and I is the identity matrix.

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 , (where A is a matrix over a field), then the eigenvalues of A satisfy the characteristic equation .

A matrix polynomial identity is a matrix polynomial equation which holds for all matricies A in a specified matrix ring

Matrix ring

In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...

M_n(R).