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

This paper presents a group of analytical formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function $\varphi (X)=(AXB+C)M(AXB+C{)}^{\ast }+D$ and use them to derive necessary and sufficient conditions for the two types of multiple quadratic matrix-valued function {align*} (, \sum_{i = 1}^{k}A_iX_iB_i + C ,)M(,\sum_{i = 1}^{k}A_iX_iB_i + C ,)^{} +D, \ \ \ \sum_{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, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},k$. L"owner partial ordering optimizations of the two matrix-valued functions are studied and their solutions are characterized.