Geodesic
On the Unitary Similarity of Matrix Families
Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 815-826

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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. 
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     author = {Yu. A. Alpin and Kh. D. Ikramov},
     title = {On the {Unitary} {Similarity} of {Matrix} {Families}},
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Yu. A. Alpin; Kh. D. Ikramov. On the Unitary Similarity of Matrix Families. Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 815-826. http://geodesic.mathdoc.fr/item/MZM_2003_74_6_a1/