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

This paper studies the quadratic matrix-valued function phi(X) = DXAX*D* + DXB B*X*D* + C 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 phi(X) = 0 to have a solution, as well as the four matrix inequalities phi(X) > (>=, <, <=) 0 in the Lowner partial ordering to be feasible, respectively; (ii) necessary and sufficient conditions for the four matrix inequalities phi(X) > (>=, <, <=) 0 in the Lowner partial ordering to hold for all matrices X, respectively; (iii) the two matrices <(X)over cap> and (X) over tilde such that the inequalities phi(X) >= phi((X) over cap) and phi(X) <= phi((X) over tilde) hold for all matrices X in the Lowner partial ordering, respectively. An application of the quadratic matrix-valued function in control theory is also presented. (C) 2012 Elsevier Inc. All rights reserved.