Words and normality of matrices
Bo-Ying Wang, Fuzhen Zhang · Linear and Multilinear Algebra · 1995
Let A’ denote the conjugate transpose of an n×n complex matrix A and let (A,A) be a word in A and A′ wilh length m The following are shown: 1.If (A, A *) or its cycle contains A2 or (A *)2 and if tr(A,A *)=tr(A * A) m/2 then A is a normal matrix; 2.If the difference of the numbers of A's and A* 's in the word is k≠0, then tr (A *) = tr(A * A)m/2 if and only if A k = (A *A) k/2. A number of consequences are also presented.