Solutions of Polynomials over Matrices

Abstract

If we are given an Nth degree polynomial over the complex numbers, we know that it has exactly n solutions. However, this is not true for an Nth degree polynomial over matrices. The reason lies in the many differences between matrices, which are a ring, and complex numbers, which are a field. The question then becomes: How many solutions exist? Researchers have developed a geometric approach to solving this problem, which shows the maximum number of diagonalizable solutions for an Nth degree polynomial over k x k matrices. This approach also leads to results that demonstrate which numbers of diagonalizable solutions are possible.