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Assume that $A_{1},...,A_{s}$ are complex $n\times n$ matrices. We give acomputable criterion for existence of a common eigenvector of $A_{i}$ whichgeneralize the result of D. Shemesh established for two matrices. We use thiscriterion to prove some necessary and sufficient condition for $A_{i}$ to havea common invariant subspace of dimension $d$, $2\leq d<n$, if every $A_{i}$ haspairwise different eigenvalues. Finally, we observe that the set of allmatrices having multiple eigevalues has Lebesgue measure 0 and thus thecondition is sufficient in practical applications. Being motivated by quantuminformation theory, we give a flavour of such applications for irreduciblecompletely positive superoperators.

Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators