A Note on the Convexity of $\log \det (I+K{X}^{-1})$ and its Constrained Optimization Representation

This note provides another proof for the {\em convexity} ({\em strict convexity}) of $\log \det (I+K{X}^{-1})$ over the positive definite cone for any given positive semidefinite matrix $K⪰0$ (positive definite matrix $K\succ 0$) and the {\em strictly convexity} of $\log \det (K+X^{-1})$ over the positive definite cone for any given $K⪰0$. Equivalent optimization representation with linear matrix inequalities (LMIs) for the functions $\log \det ( I + KX^{-1} )$and$\log \det (K + X^{-1})$ are presented. Their optimization representations with LMI constraints can be particularly useful for some related synthetic design problems.