摘要： For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors xj∈Cn such that the scalar products 〈Axj,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 (n≥2) nonnegative matrix with the permutationally irreducible real part of the form[0A100⋱⋱Am−100],where m≥3 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). 出版者： Elsevier Inc 出版日期： 2015-03-15 出處： Linear algebra and its applications, 2015-03, Vol.469, p.594-608 資源來源： Elsevier ScienceDirect 版權： 2014 Elsevier Inc. 識別號： ISSN: 0024-3795 識別號： EISSN: 1873-1856 識別號： DOI: 10.1016/j.laa.2014.12.003
