矩阵谱序的Grassmann公式

中文摘要 AI 辅助

对于Hermitian矩阵$A$和$B$，我们证明$(A\join B)\oplus(A\meet B)$与$A\oplus B$酉等价，其中join和meet是在Olson谱序中取的。取迹回答了Bourin和Lee的一个问题：对于半正定矩阵，$\Tr(A\join B)=\Tr(A+B)$当且仅当$A\meet B=0$。迭代直和恒等式给出了有限个正矩阵的迹公式。谱上确界的迹等于和的迹，当且仅当值域构成代数直和。对于每个固定的$1<p<\infty$，谱上确界与和具有相等的Schatten $p$-范数，当且仅当值域两两正交。最后，我们建立了Frobenius内积与谱序之间的一个优雅公式。

英文摘要

For Hermitian matrices $A$ and $B$, we prove that $(A\join B)\oplus(A\meet B)$ is unitarily equivalent to $A\oplus B$, where the join and meet are taken in Olson's spectral order. Taking traces answers a question of Bourin and Lee: for positive semidefinite matrices, $\Tr(A\join B)=\Tr(A+B)$ if and only if $A\meet B=0$. Iterating the direct-sum identity gives a trace formula for a finite family of positive matrices. The trace of the spectral supremum equals the trace of the sum precisely when the ranges form an algebraic direct sum. For each fixed $1<p<\infty$, the spectral supremum and the sum have equal Schatten $p$-norms if and only if the ranges are pairwise orthogonal. Finally we establish an elegant formula for the Frobenius inner product and the spectral order.

发表机构

• University of Monastir（蒙纳斯蒂尔大学）
• Kyungpook National University（庆北国立大学）

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