Mathematical Notes

, Volume 87, Issue 5, pp 903–907

A short note on the Frobenius norm of the commutator


DOI: 10.1134/S0001434610050305

Cite this article as:
Wu, YD. & Liu, XQ. Math Notes (2010) 87: 903. doi:10.1134/S0001434610050305


This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ℝn×n. We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XYXYF2 ≤ 2‖XF2YF2 is finally improved into
$$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$
. In addition, a further improvement is made.

Key words

commutator Frobenius norm Böttcher and Wenzel’s conjecture random matrix 

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Huaiyin Institute of TechnologyHuaiyinChina

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