Unitarily achievable zero patterns and traces of words in \(A\) and \(A\)
As a main objective, the authors investigate the problem of which patterns of zeros are achievable in matrices via unitary similarity of complex square matrices. A further objective is to bring to the attention of the linear algebra community some recent advances concerning the old problem of unitary similarity of two complex \(n\times n\) matrices \(A\) and \(B\). A theorem due to \textit{W. Specht} [Jahresber. Dtsch. Math.-Ver. 50, Abt. 1, 19--23 (1940; Zbl 0023.00203)] gives the N\&S condition: \(\text{tr}(w(A,A^*))= \text{tr}(w(B,B^*))\) for all words \(w\) in two non-commuting variables. This was later improved in that the condition need only hold for all words of length at most \(2n^2\) or even \(n^2\).
Specht's criterion for unitary similarity
scientific article; zbMATH DE number 5123957
scientific article