The quadratic equation X2 + AX + B = 0, where the coefficients A, B and the unknown X are all 2×2 matrices over ℂ, has some surprising properties. For example, it is possible to choose matrices A and B so that this equation has exactly 0, 1, 2, 3, 4, 5, or 6 solutions, but if there are more than 6 solutions there must be infinitely many solutions. This result is described at http://www.math.rutgers.edu/~rwilson/polynomial_equations.pdf
It would be interesting to know the corresponding result for an equation of degree m where the coefficients and unknowns are n×n matrices over the complex numbers. A procedure for describing all solutions of such an equation is known and it is known that it is possible to choose coefficients so that the number of solutions is exactly equal to the binomial coefficient "mn choose n". It may well be that any number of solutions between 0 and "mn choose n" is possible but that no larger finite number of solutions is possible, but this has not been proved. The proposed project is to find all possible numbers of solutions for some collection of values of m and n. (It would be nice to do this for all m and n, but that might be too hard. One would probably start with the case of a cubic equation in 2 by 2 matrices, and then proceed to other cases if possible.) This problem can be investigated using standard tools of linear algebra (in particular, rational canonical form).
(description from Prof. Wilson)For any n and any m < n(n-1)/2, there is an nth degree equation over 2×2 matrices with exactly m solutions
To see my pdfs and such, e-mail me and I'll send you the links.
Now to try to answer the question, "is there an nth degree polynomial over k×k matrices such that there are exactly m solutions?"
Here are some interesting quadratic equations over 3×3 matrices
Also,
This page last updated 2006 July 24