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

Abstract

We establish a log-majorization inequality comparing the eigenvalues of the interlaced product $Y^t{X}^{\ast }{Y}^{1-t}X$ with those of $X^{\ast }YX$, valid for every positive semi-definite $Y$ and every normal $X$, with the inequality reversing for $t\notin [0,1]$ 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 $$\det (A^{\ast }A+|BA{|}^p)\le \det (A{A}^{\ast }+|A^{\ast }{B}^{\ast }{|}^p)$$ for arbitrary $A$, normal $B$ and $p\ge 0$, and we give a complete eigenvalue picture for products of weighted geometric means, sharpening and complementing a theorem of Hiai and Lin.