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Formulas for calculating the extremum ranks and inertias of a four-term quadratic

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



Entry Submitted: 02/26/2012
Entry Accepted: 02/28/2012
Entry Last Modified: 05/26/2012

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