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Let $r$ be a nonconstant noncommutative rational function in $m$ variablesover 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 andmultiplicative Waring problems for matrix algebras. Concerning the linearproblem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can bewritten as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variationsof 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, everynonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\inM_n(K)^m$.

Matrix evaluations of noncommutative rational functions and Waring problems