Gau-Wu numbers of nonnegative matrices

For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors ^xjCn such that the scalar products (A^xj,^xj) lie on the boundary W(A) of the numerical range W(A). This number is called the Gau-Wu number of the matrix A. If A is an n-by-n (n2) nonnegative matrix with the permutationally irreducible real part of the form[0^A100Am-₁00],where m3 and the diagonal zeros are zero square matrices, then k(A) has an upper bound m-1. In addition, we also obtain necessary and sufficient conditions for k(A)=m-1 for such a matrix A. Another class of nonnegative matrices we study is the doubly stochastic ones. We prove that the value of k(A) is equal to 3 for any 3-by-3 doubly stochastic matrix A. For any 4-by-4 doubly stochastic matrix, we also determine its numerical range, which is then applied to find its Gau-Wu numbers. Furthermore, a lower bound of the Gau-Wu number k(A) is also found for a general n-by-n (n>5) doubly stochastic matrix A via the possible shapes of W(A).