Closed-form formulas for calculating the extremal ranks and inertias of a quadratic matrix-valued function and their applications

2013/01/11 by Tian, Yongge
#15A24 #15B57 #65K10 #90C20 #90C22 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1301.2567

Abstract

This paper presents a group of analytical formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function ϕ(X) = ( AXB + C )M( AXB + C)^* + D and use them to derive necessary and sufficient conditions for the two types of multiple quadratic matrix-valued function align* ( ∑_{i = 1}^kA_iX_iB_i + C )M( ∑_{i = 1}^kA_iX_iB_i + C )^* +D, ∑_{i = 1}^k( A_iX_iB_i + C_i )M_i( A_iX_iB_i + C_i )^* +D align* to be semi-definite, respectively, where A_i, B_i, C_i, C, D, M_i and M are given matrices with M_i, M and D Hermitian, i =1,..., k. Löwner partial ordering optimizations of the two matrix-valued functions are studied and their solutions are characterized.

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