AI 中文总结
研究建立正规矩阵对数优超不等式，扩展厄米特结果。通过该不等式解决行列式猜想正规矩阵扩展，给出加权几何均值乘积特征值图景，完善相关定理。
AI 中文摘要
我们建立了一个对数优超不等式，用于比较交错积\(Y^t X^*Y^{1 - t}X\)与\(X^*YX\)的特征值，该不等式对每个半正定\(Y\)和每个正规\(X\)都成立。当\(Y\)正定且\(t \notin[0,1]\)时不等式反向。这将已知的厄米特结果扩展到了更大的正规矩阵类，证明了正规性是精确的结构假设。作为应用，解决了林的一个行列式猜想的正规矩阵扩展，给出了加权几何均值乘积的完整特征值图景。
英文摘要
We establish a log-majorization inequality comparing the eigenvalues of the interlaced product $Y^t X^*Y^{1-t}X$ with those of $X^*YX$, 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 $$\det(A^*A + |BA|^p) \le \det(AA^* + |A^*B^*|^p)$$ 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.
