Words and Normality of Matrices

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 A ² 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.