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 functionϕ(X)=DXAX∗D∗+DXB+B∗X∗D∗+Cϕ(X)=DXAX∗D∗+DXB+B∗X∗D∗+Cthrough 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 ϕ(X)=0ϕ(X)=0 to have a solution, as well as the four matrix inequalities ϕ(X)>(⩾,<,⩽)0 in the Löwner partial ordering to be feasible, respectively;(ii)necessary and sufficient conditions for the four matrix inequalities ϕ(X)>(⩾,<,⩽)0 in the Löwner partial ordering to hold for all matrices X, respectively;(iii)the two matrices X^ and X∼ such that the inequalities ϕ(X)⩾ϕ(X^) and ϕ(X)⩽ϕ(X∼) hold for all matrices X in the Löwner partial ordering, respectively.An application of the quadratic matrix-valued function in control theory is also presented.