On the Gau-Wu number for some classes of matrices

For a given n× n matrix A, let k(A) stand for the maximal number of orthonormal vectors x^j such that the scalar products «Ax ^j,x^j» lie on the boundary of the numerical range W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau-Wu number of the matrix A. We compute k(A) for two classes of n × n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)≥2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau-Wu number. Finally, matrices with maximal Gau-Wu number (k(A)=n) are considered.