A log-majorization inequality for normal matrices with applications to determinantal inequalities and geometric means

2026/07/23 by Mohammad Mahdi Ghabries
Mathematics · #math.FA

paper · pdf

Abstract

We establish a log-majorization inequality comparing the eigenvalues of the interlaced product Y^t X^*Y^1-tX with those of X^*YX, valid for every positive semi-definite Y and every normal X, with the inequality reversing for t ∉[0,1] when Y is positive definite. This extends known Hermitian results to the strictly larger class of normal matrices, where normality is shown to be the exact structural hypothesis, not a technical convenience. A counterexample proves the result can fail without it. As applications, we settle a normal-matrix extension of a determinantal conjecture of Lin, proving det(A^*A + |BA|^p) ≤ det(AA^* + |A^*B^*|^p) for arbitrary A, normal B and p ≥ 0, and we give a complete eigenvalue picture for products of weighted geometric means, sharpening and complementing a theorem of Hiai and Lin.

Related

• Some matrix inequalities of log-majorization type
• Weak log majorization and determinantal inequalities
• New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality
• A generalization of Araki's log-majorization
• Weak log-majorization between the geometric and Wasserstein means