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#### Keywords

Matrix Inequality, Unitarily Invariant Norm, Positive semidefinite matrix

#### 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.

#### Recommended Citation

Audenaert, Koenraad MR.
(2015),
"A norm inequality for pairs of commuting positive semidefinite matrices",
*Electronic Journal of Linear Algebra*,
Volume 30.

DOI: http://dx.doi.org/10.13001/1081-3810.2829