ELA, Volume 3, pp. 142-152, August 1998, abstract.

On Spaces of Matrices with a Bounded Number of Eigenvalues

Raphael Loewy and Nizar Radwan

The following problem, originally proposed by Omladic and  Semrl 
[Linear Algebra Appl., 249:29--46 (1996)], is considered. Let k and 
n be positive integers such that k<n. Let L be a subspace of M_n(F), 
the space of n by n matrices over a field F, such that each A in L 
has at most k distinct eigenvalues (in the algebraic closure of F). 
Then, what is the maximal dimension of L. Omladic and Semrl assumed 
that F is the field of complex numbers, and solved the problem for 
k=1, k=2 and n odd, and k=n-1 (under a mild assumption). In this 
paper, their results for k=1 and k=n-1 are extended to any F such that 
char(F)=0, and a solution for k=2 and any n, and for k=3 is given.