Weak log majorization and determinantal inequalities

TL;DR

This paper establishes a weak log majorization inequality for eigenvalues of certain block matrices derived from positive definite matrices, generalizes a known determinantal inequality, and explores related inequalities and open questions.

Contribution

It introduces a new weak log majorization result for eigenvalues of block matrices, extending previous determinantal inequalities and highlighting differences with singular value inequalities.

Findings

Proves a weak log majorization inequality for eigenvalues of block matrices.

Shows that the eigenvalue inequality does not hold for singular values.

Provides a generalization of Matic's determinantal inequality and discusses related open questions.

Abstract

Denote by $\P_n$ the set of $n\times n$ positive definite matrices. Let $D = D_1\oplus \dots \oplus D_k$, where $D_1\in \P_{n_1}, \dots, D_k \in \P_{n_k}$ with $n_1+\cdots + n_k=n$. Partition $C\in \P_n$ 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*} \lambda (C^{-1}_1D_1\oplus \cdots \oplus C^{-1}_kD_k)\prec_{w \,\log} \lambda(C^{-1}D), \end{equation*} where $\lambda(A)$ 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.