TY - JOUR

T1 - Gau-Wu numbers of nonnegative matrices

AU - Lee, Hsin Yi

N1 - Publisher Copyright:
© 2014 Elsevier Inc. Allrightsreserved.

PY - 2015

Y1 - 2015

N2 - For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors xjCn 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 (n2) nonnegative matrix with the permutationally irreducible real part of the form[0A100Am-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).

AB - For any n-by-n matrix A, we consider the maximum number k=k(A) of orthonormal vectors xjCn 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 (n2) nonnegative matrix with the permutationally irreducible real part of the form[0A100Am-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).

KW - Doubly stochastic matrix

KW - Nonnegative matrix

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=84937147783&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2014.12.003

DO - 10.1016/j.laa.2014.12.003

M3 - 期刊論文

AN - SCOPUS:84937147783

SN - 0024-3795

VL - 469

SP - 594

EP - 608

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1

ER -