Polynomials with coefficients over Fields (Real or Complex numbers) are commonly seen in many applications, but rare are the Polynomial Matrix (PM) or Polynomials with coefficient over (non-abelian Ring) Matrix.
For curiosity, I google on PM, however there isn't any information on its application except the definition in: https://en.m.wikipedia.org/wiki/Polynomial_matrix
I would like to know in what areas, situations, problems, in which we use PM ?
Appreciate anyone who could enlighten me.
Thanks in advance.
[Note] below the difference between Polynomial Matrix (PM) & Matrix Polynomial(MP) :
Let Aj (j = 0...n) be any square matrix. I be the unity matrix.
Polynomial Matrix (PM) : Eg. X² + Aj. X + Aj
Or
Matrix Polynomial (MP): Eg. (Aj)²+ 3.Aj + I
MP is just a combination of matrices Aj. We can apply Caley-Hamilton Theorem to simplify its computation.
PM is polynomial with coefficient in matrices Aj (j = 0...n).
There are several polynomial matrix equations, for example the Sylvester Equation, the Algebraic Riccati Equation, or Fermat's equation and others. They have applications for various fields of mathematics and physics. There is a book book "Matrix Polynomials", of which the authors are I. Gohberg, P. Lancaster and L. Rodman.
Also on this site we find many posts asking about polynomial matrix equations:
Solving polynomial matrix equations over finite fields
Solving a simple matrix polynomial
Matrix Equation Simplification