Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications

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• 作者：Yongge Tian ^{yongge.tian@gmail.com}
• 关键词：15A09 ; 15A24 ; 15A63 ; 15B57 ; 65K10 ; 90C20 ; 90C22
• 刊名：Linear Algebra and its Applications
• 出版年：2012
• 期刊代码：38_00243795
• 类别：mt
• 出版时间：1 August, 2012
• 卷：437
• 期：3
• 页码：835-859
• 文件大小：423 K

摘要

This paper studies the quadratic matrix-valued function

through some expansion formulas for ranks and inertias of Hermitian matrices, where A, B, C and D are given complex matrices with A and C Hermitian, X is a variable matrix, and denotes the conjugate transpose of a complex matrix. We first introduce an algebraic linearization method for studying this matrix-valued function, and establish a group of explicit formulas for calculating the global maximum and minimum ranks and inertias of this matrix-valued function with respect to the variable matrix X. We then use these rank and inertia formulas to derive:

(i)

necessary and sufficient conditions for the matrix equation to have a solution, as well as the four matrix inequalities in the L枚wner partial ordering to be feasible, respectively;

(ii)

necessary and sufficient conditions for the four matrix inequalities in the L枚wner partial ordering to hold for all matrices X, respectively;

(iii)

the two matrices and such that the inequalities and hold for all matrices X in the L枚wner partial ordering, respectively.

An application of the quadratic matrix-valued function in control theory is also presented.