Weak log majorization and determinantal inequalities

Denote by $¶_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in ¶_{n_1}, \dots, D_k \in ¶_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in ¶_n$ according to $(n_1, \dots, n_k)$ so that $\Diag C = C_1\oplus \dots \oplus C_k$. We prove the following weak log majorization result: \begin{equation*} λ(C^{-1}_1D_1\oplus \cdots \oplus C^{-1}_kD_k)\prec_{w \,\log} λ(C^{-1}D), \end{equation*} where $λ(A)$ denotes the vector of eigenvalues of $A\in \Cnn$. 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_1\oplus \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.

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