On the Gau–Wu number for some classes of matrices

For a given n×nn×n matrix A  , let k(A)k(A) stand for the maximal number of orthonormal vectors xjxj such that the scalar products 〈Axj,xj〉〈Axj,xj〉 lie on the boundary of the numerical range W(A)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)k(A) for two classes of n×nn×n matrices A  . A simple and explicit expression for k(A)k(A) for tridiagonal Toeplitz matrices A   is derived. Furthermore, we prove that k(A)=2k(A)=2 for every pure almost normal matrix A. Note that for every matrix A   we have k(A)⩾2k(A)⩾2, and for normal matrices A   we have k(A)=nk(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)=nk(A)=n) are considered.