Fillmore's theorem and sums of nilpotent quaternion matrices

Abstract

Fillmore's Theorem states that an $n\times n$ nonscalar matrix $A$ over a field is similar to a matrix whose diagonal entries are $\lambda_1,\ldots,\lambda_n$ provided that $\sum_{i=1}^n\lambda_i = \mathrm{Tr} A.$ It is shown that an analog of Fillmore's Theorem holds for matrices over a quaternion division algebra. As a consequence, a matrix that is nonscalar over the underlying field and has zero reduced trace is a sum of at most three nilpotent matrices. The upper bound three is optimal since it is shown that there are matrices that are sums of exactly three nilpotent matrices. A characterization of sums of two square-zero real quaternion matrices is also presented.