A norm inequality for pairs of commuting positive semidefinite matrices

Koenraad Audenaert

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Abstract

For $k=1,\ldots,K$, let $A_k$ and $B_k$ be positive semidefinite matrices such that, for each $k$, $A_k$ commutes with $B_k$. We show that, for any unitarily invariant norm, \[ |||\sum_{k=1}^K A_kB_k||| \le ||| (\sum_{k=1}^K A_k)\;(\sum_{k=1}^K B_k)|||. \] The $K=2$ case was recently conjectured by Hayajneh and Kittaneh and proven by them for the trace norm and the Hilbert-Schmidt norm. A simple application of this norm inequality answers a question by Bourin in the affirmative.
Original languageEnglish
Article number5
Pages (from-to)80-84
Number of pages5
JournalElectronic Journal of Linear Algebra
Volume30
DOIs
Publication statusPublished - 16 Feb 2015

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