Words and Normality of Matrices
Linear and Multilinear Algebra
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 normalmatrix; 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.
Wang, Bo-Ying and Zhang, Fuzhen, "Words and Normality of Matrices" (1995). Mathematics Faculty Articles. 133.