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 A(i) denotes a matrix of constant coefficients, and A(p) is non-zero. An example 3×3 polynomial matrix, degree 2: We can express this by saying that for a ring R, the rings M_n(R[X]) and (M_n(R))[X] 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 λI − A is the characteristic matrix of the matrix A. Its determinant, |λI − A| is the characteristic polynomial of the matrix A.