Unitary similarity of matrices with quadratic minimal polynomials

AbstractThe fact that given complex n×n matrices A and B are (or are not) unitarily similar can be verified with the help of the Specht–Pearcy criterion. Its application, however, involves a huge amount of computational work; to get a positive answer, one should compare the traces of all the products composed of A and A∗ with length up to 2n2 with the traces of similar products composed of B and B∗. For some matrices A, B, most of this work is redundant. For instance, when A and B are normal matrices, they only need to have identical eigenvalues to be unitarily similar, and this condition can be verified by comparing two n-tuples of traces. In this paper, we identify another class of matrices where unitary similarity among its members can be economically verified. These are matrices with quadratic minimal polynomials. If A and B are matrices of this kind, then, to be unitarily similar, they need to have the same eigenvalues and the same singular values, which can be verified by comparing two 2n-tuples of traces. Two widely known subclasses of matrices with quadratic minimal polynomials are projectors and involutions. For these subclasses, we give yet another derivation of the unitary similarity criterion above, based on the canonical form for oblique projectors found by D. Djoković