  Assume that $A_{1},...,A_{s}$ are complex $n\times n$ matrices. We give a
computable criterion for existence of a common eigenvector of $A_{i}$ which
generalize the result of D. Shemesh established for two matrices. We use this
criterion to prove some necessary and sufficient condition for $A_{i}$ to have
a common invariant subspace of dimension $d$, $2\leq d<n$, if every $A_{i}$ has
pairwise different eigenvalues. Finally, we observe that the set of all
matrices having multiple eigevalues has Lebesgue measure 0 and thus the
condition is sufficient in practical applications. Being motivated by quantum
information theory, we give a flavour of such applications for irreducible
completely positive superoperators.
