- Formulas for calculating the extremum ranks and inertias of a four-term quadratic Tian Yongge (yongge.tiangmail.com) Abstract: 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 $(\cdot)^*$ 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: \begin{enumerate} \item[{\rm (i)}] necessary and sufficient conditions for the matrix equation $\phi(X) = 0$ to have a solution, as well as the four matrix inequalities $\phi(X) > (\geqslant, \ <, \ \leqslant)\, 0$ in the L\"owner partial ordering to be feasible, respectively; \item[{\rm (ii)}] necessary and sufficient conditions for the four matrix inequalities $\phi(X) > (\geqslant, \ <, \ \leqslant)\, 0$ in the L\"owner partial ordering to hold for all $X$, respectively; \item[{\rm (iii)}] the two matrices $\widehat{X}$ and $\widetilde{X}$ such that the inequalities $\phi(X) \geqslant \phi(\widehat{X})$ and $\phi(X) \leqslant \phi(\widetilde{X})$ hold for any matrix $X$ in the L\"owner partial ordering, respectively. \end{enumerate} An application of the quadratic matrix-valued function in control theory is also presented. Keywords: quadratic matrix-valued function; generalized algebraic Riccati equation; matrix inequality; rank; inertia; L\"owner partial ordering; optimization; linearization method Category 1: Global Optimization Category 2: Nonlinear Optimization Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Citation: Download: Entry Submitted: 02/26/2012Entry Accepted: 02/28/2012Entry Last Modified: 05/26/2012Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.