PreprintCC BY 4.0arXiv
A Grassmann formula for the spectral order of matrices
Abstract
For Hermitian matrices A and B, we prove that (Ajoin B)⊕(Ameet B) is unitarily equivalent to A⊕ 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(Ajoin B)=Tr(A+B) if and only if Ameet 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<∞, 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.
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BibTeX
@misc{aouichaoui2026grassmann,
title = {{A Grassmann formula for the spectral order of matrices}},
author = {Mohamed Amine Aouichaoui and Eun-Young Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39659},
eprint = {2609.39659},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39659},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Grassmann formula for the spectral order of matrices
AU - Mohamed Amine Aouichaoui
AU - Eun-Young Lee
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39659
UR - https://arxiv.org/abs/2609.39659
AB - For Hermitian matrices A and B, we prove that (Ajoin B)⊕(Ameet B) is unitarily equivalent to A⊕ 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(Ajoin B)=Tr(A+B) if and only if Ameet 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<∞, 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.
ER -