  We obtain a nontrivial bound on the number of solutions to the equation $$
A^{x_1} + \ldots + A^{x_\nu} = A^{x_{\nu+1}} + \ldots + A^{x_{2\nu}}, \quad 1
\le x_1, \ldots,x_{2\nu} \le \tau, $$ with a fixed $n\times n$ matrix $A$ over
a finite field $\mathbb F_q$ of $q$ elements of multiplicative order $\tau$. We
give applications of our result to obtaining a new bound of additive character
sums with a matrix exponential function, which is nontrivial beyond the
square-root threshold. For $n=2$ this equation has been considered by Kurlberg
and Rudnick (2001) (for $\nu=2$) and Bourgain (2005) (for large $\nu$) in their
study of quantum ergodicity for linear maps over residue rings. Here we use a
new approach to improve their results. We also obtain a bound on Kloosterman
sums over small subgroups, of size below the square-root threshold.
