Gau–Wu numbers of nonnegative matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)k=k(A) of orthonormal vectors xj∈Cnxj∈Cn such that the scalar products 〈Axj,xj〉〈Axj,xj〉 lie on the boundary ∂W(A)∂W(A) of the numerical range W(A)W(A). This number is called the Gau–Wu number of the matrix A. If A is an n-by-n   (n≥2n≥2) nonnegative matrix with the permutationally irreducible real part of the form[0A100⋱⋱Am−100],where m≥3m≥3 and the diagonal zeros are zero square matrices, then k(A)k(A) has an upper bound m−1m−1. In addition, we also obtain necessary and sufficient conditions for k(A)=m−1k(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)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)k(A) is also found for a general n-by-n   (n≥5n≥5) doubly stochastic matrix A   via the possible shapes of W(A)W(A).