Polynomial matrix

Template:Distinguish 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:

P = \sum_{n = 0}^{p} A (n) x^{n} = A (0) + A (1) x + A (2) x^{2} + \dots + A (p) x^{p}

where $A (i)$ denotes a matrix of constant coefficients, and $A (p)$ is non-zero. An example 3×3 polynomial matrix, degree 2:

P = (\begin{matrix} 1 & x^{2} & x \\ 0 & 2 x & 2 \\ 3 x + 2 & x^{2} - 1 & 0 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 2 & - 1 & 0 \end{matrix}) + (\begin{matrix} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{matrix}) x + (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) x^{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

A polynomial matrix over a field with determinant equal to a non-zero element of that field is called unimodular, and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials are polynomials of degree 0 – nonzero constants, because an inverse of an arbitrary polynomial of higher degree is a rational function.
The roots of a polynomial matrix over the complex numbers are the points in the complex plane where the matrix loses rank.
The determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients.^[1]

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.