  Let $r$ be a nonconstant noncommutative rational function in $m$ variables
over an algebraically closed field $K$ of characteristic 0. We show that for
$n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$
distinct and nonzero eigenvalues. This result is used to study the linear and
multiplicative Waring problems for matrix algebras. Concerning the linear
problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be
written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations
of this result for the case where $r$ is a noncommutative polynomial.
Concerning the multiplicative problem, we show, among other results, that if
$f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every
nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in
M_n(K)^m$.
