Polynomial matrix

In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices.
A univariate polynomial matrix "P" of degree "p" is defined as:
where formula_2 denotes a matrix of constant coefficients, and formula_3 is non-zero. 
An example 3×3 polynomial matrix, degree 2:
We can express this by saying that for a ring "R", the rings formula_5 and
formula_6 are isomorphic.
Properties.
Note that polynomial matrices are "not" to be confused with monomial matrices, which are simply matrices with exactly one non-zero entry in each row and column.
If by λ we denote any element of the field over which we constructed the matrix, by "I" the identity matrix, and we let "A" be a polynomial matrix, then the matrix λ"I" − "A" is the characteristic matrix of the matrix "A". Its determinant, |λ"I" − "A"| is the characteristic polynomial of the matrix "A".