Polynomial matrix

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A polynomial matrix or matrix polynomial is a matrix whose elements are polynomials. Univariate or multivariate.

A univariate polynomial matrix P of degree p is defined as:

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

where $A (i)$ denotes a matrix of constant coefficients, and $A (p)$ is non-zero. Thus a polynomial matrix is the matrix-equivalent of a polynomial, with each element of the matrix satisfying the definition of a polynomial of degree p.

An example 3×3 polynomial matrix, degree 2:

$P=\begin{pmatrix} 1 & x^2 & x \\ 0 & 2x & 2 \\ 3x+2 & x^2-1 & 0 \end{pmatrix} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 2 & -1 & 0 \end{pmatrix} +\begin{pmatrix} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{pmatrix}x+\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}x^2$

We can express this by saying that for a ring R, the rings $M n (R [X])$ and $(M n (R))[X]$ are isomorphic.

[edit] Properties

A polynomial matrix over a field with determinant equal to a non-zero constant is called unimodular, and has an inverse, which 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.

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.