Hereditary Classes of Operators and Matrices
Scott McCullough, Leiba Rodman · American Mathematical Monthly · 1997
implies that the matrix A is hermitian. Here A* denotes the conjugate transpose of the matrix A. The answer is yes (of course, only the case of singular A is nontrivial), as can be proved by elementary methods: By Schur's triangularization theorem, which asserts that every square size complex matrix is unitarily similar to an upper triangular matrix [17, Theorem 2.3.1], we may assume that A itself is upper triangular. Inspection of the diagonal entries of A*A = A2 shows that all the diagonal entries of A are real and all the off-diagonal entries are zero. Thus, A is Hermitian, since it is unitarily similar to a real diagonal matrix. Consider now a symmetrizedversion of (1.1) in which the squares of both A and A* appear symmetrically: