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

This note provides another proof for the convexity (strict convexity) of log det(I + KX) over the positive definite cone for any given positive semidefinite matrix K 0 (positive definite matrix K ≻ 0) and the strictly convexity of log det(K + X) 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) and log det(K + X) are presented. Their optimization representations with LMI constraints can be particularly useful for some related synthetic design problems.