roots of polynomial with matrix coefficients Are somewhere existed a method for solving a linear equations over matrices?
For example, I have a task that is similar to next:
find $l \times l$ - matrix $A \in M_{l \times l}(\mathbf{F}_q)$ over $\mathbf{F}_q$ that vanishes $B_0 + B_1 X + B_2 X^2 + B_3 X^3 + ... + B_m X^m$ where $B_i \in M_{l \times l}(\mathbf{F}_q)$.
I have found nothing about this by Google.
Thank you!
 A: Yes, there a number of numerical methods for finding the solvents of a polynomial with matrix coefficients. Dennis, Traub, and Weber in this article give some of the relevant theory, as well as some algorithms for finding so-called "dominant solvents" for your matrix polynomial (see also their earlier article). This article presents the use of Newton's method for solving matrix polynomials.
As a tiny aside on the "companion" formulation mentioned by Federico: make sure that whatever software you're using is the version that uses the QZ algorithm, without the preliminary application of the inverse of the leading coefficient to "monicize" the matrix polynomial. It can happen that the leading coefficient is an ill-conditioned matrix, and the "monicization" leads to a degradation of accuracy in the solvents returned.
A: I'd like to provide a reference for this question. You can see the book "Matrix Polynomials", of which the authors are I. Gohberg, P. Lancaster and L. Rodman. I hope it will be helpful.
A: If you are looking for information about roots of polynomials with matrix coefficients try checking out this set of notes created by Robert L. Wilson (at Rutgers University):
http://www.math.rutgers.edu/~rwilson/polynomial_equations.pdf
Wilson has mentored a series of undergraduates who have made some small contributions to this area. Try searching for something like "DIMACS REU polynomials with matrix coefficients" and you should be able to dig up a few project summaries.
A: You can consider this as a system of $l^2$ polynomial equations in the $l^2$ matrix entries of $X$.  I doubt that you can do much better in general.  Even the quadratic case is not trivial.  For $2 \times 2$ matrices that is likely to "reduce" to solving a polynomial of degree 6 in one variable.
A: It's essentially finding $l$ solutions of a polynomial eigenvalue problem (i.e., find a vector $v$ s.t. $(\sum x^i B_i)v=0$. Existing methods to solve it are usually based on linearization (a matrix version of the "companion matrix"); Matlab has a function polyeig that does all the work for you. Check Chapter 9 of Templates for the solution of algebraic eigenvalue problems by Zhaojun Bai.
In alternative, there are iterative methods to solve directly the matrix equation, such as Cyclic Reduction for the quadratic case - you find something in Numerical methods for structured Markov chains, Bini, Latouche, Meini, and in the article Solving matrix polynomial equations arising in queueing problems by the same authors.
A: see
International Mathematical Forum, 3, 2008,  no.  17,  829  - 836
A  Note on  a Generalization of an  Extension of the  Cayley-Hamilton Theorem
this will reply your request
best
issam
