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On global optimizations of the rank and inertia of the matrix function $A_1- B_1XB^*_1$ subject to a pair of matrix equations $[\,B_2XB^*_2, \, B_3XB^*_3 \,] = [\,A_2, \, A_3\,]$

Yongge Tian(yongge.tian***at***gmail.com)

Abstract: For a given linear matrix function $A_1 - B_1XB^*_1$, where $X$ is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations $B_2XB^*_2 = A_2$ and $B_3XB_3^* = A_3$. As applications, we give necessary and sufficient conditions for the triple matrix equations $B_1XB^*_1 =A_1$, $B_2XB^*_2 = A_2$ and $B_3XB^*_3 = A_3$ to have a common Hermitian solution. In addition, we discuss the global optimizations on the rank and inertia of the common Hermitian solution of the pair of matrix equations $B_2XB^*_2 = A_2$ and $B_3XB^*_3 = A_3$.

Keywords: Matrix function; matrix equation; common Hermitian solution; rank; inertia; maximization; minimization; L\"owner partial ordering

Category 1: Global Optimization (Theory )

Citation:

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Entry Submitted: 04/10/2011
Entry Accepted: 04/10/2011
Entry Last Modified: 04/10/2011

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