Denote by $P_n$ the set of $ntimes n$ positive definite matrices. Let $D = D_1oplus dots oplus D_k$, where $D_1in P_{n_1}, dots, D_k in P_{n_k}$ with $n_1+cdots + n_k=n$. Partition $Cin P_n$ according to $(n_1, dots, n_k)$ so that $Diag C = C_1oplus dots oplus C_k$. We prove the following weak log majorization result: begin{equation*} lambda (C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} lambda(C^{-1}D), end{equation*} where $lambda(A)$ denotes the vector of eigenvalues of $Ain Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., begin{equation*} s(C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} s(C^{-1}D) end{equation*} is not true. As an application, we provide a generalization of a determinantal inequality of Matic cite[Theorem 1.1]{M}. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi cite[Theorem 2]{C} and give a weak log majorization open question.
Download