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A square matrix is nonderogatory if its Jordan blocks have distincteigenvalues. We give canonical forms (i) for nonderogatory complex matrices upto unitary similarity and (ii) for pairs of complex matrices up to similarity,in which one matrix has distinct eigenvalues. The types of these canonicalforms are given by undirected and, respectively, directed graphs with noundirected cycles.

A canonical form for nonderogatory matrices under unitary similarity