On the Unitary Similarity of Matrix Families

The classical Specht criterion for the unitary similarity between two complex $(n\times n)$ matrices is extended to the unitary similarity between two normal matrix sets of cardinality $m$. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht"s theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht–Pearcy criterion.