半正定分块矩阵奇异值不等式的研究  

Singular value inequalities of positive semidefinite block matrices

作　　者：崔姝姝 任芳国[1] CUI Shu-shu;REN Fang-guo(School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, Shaanxi, China)

机构地区：[1]陕西师范大学数学与统计学院,陕西西安710062

出　　处：《宝鸡文理学院学报（自然科学版）》2021年第3期8-13,19,共7页Journal of Baoji University of Arts and Sciences(Natural Science Edition)

基　　金：国家自然科学基金项目(11471200)。

摘　　要：目的研究2×2半正定分块矩阵的奇异值不等式。方法通过半正定矩阵的性质、特征值和奇异值之间的关系及Courant-Fischer定理进行研究。结果在一定的条件下,得到了2×2半正定分块矩阵及其在单调递增的凸函数上的奇异值不等式,以及一些特殊的半正定分块矩阵和单调递增的凸函数奇异值的特殊情况。结论完善了矩阵奇异值不等式理论,促进矩阵不等式理论的发展。Purposes—To research the singular value inequalities for the 2×2 positive semidefinite block matrices.Methods—The research is conducted via the properties of positive semidefinite matrices,and the relationship between the eigenvalue and singular values,and Courant-Fischer max-min theorem.Result—The singular value inequalities of the 2×2 positive semidefinite block matrices are obtained together with its monotonic increasing convex function under certain conditions.Then the special case of some special positive semidefinite block matrices and monotonic increasing convex functions are given.Conclusion—The theory of the singular value inequalities of matrices is perfected,which facilitates the development of the theory of matrix inequalities.

关 键 词：半正定分块矩阵 凸函数 奇异值不等式 Courant-Fischer定理 

分 类 号：O151.21[理学—数学]