Unitary similarity of matrices with quadratic minimal polynomials
The fact that given complex \(n\times n\) matrices \(A\) and \(B\) are (or are not) unitarily similar can be verified with the help of the Specht-Pearcy criterion [cf. \textit{C. Pearcy}, Pacific J. Math. 12, 1405-1416 (1962; Zbl 0144.37801)]. Its application involves a huge amount of computational work. For some matrices \(A,B\), most of this work is redundant. The paper identifies a 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, the authors give yet another derivation of the unitary similarity criterion above, based on the canonical form for oblique projectors found by \textit{D. Ž. Doković} [Aequationes Math. 42, No. 2/3, 220-224 (1991; Zbl 0749.15005)].
scientific article; zbMATH DE number 1790240