, Volume 87, Issue 5-6, pp 903-907

A short note on the Frobenius norm of the commutator

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Abstract

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 ℝ n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XYXY F 2 ≤ 2‖X F 2 Y F 2 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.

Published in Russian in Matematicheskie Zametki, 2010, Vol. 87, No. 6, pp. 935–940.
The text was submitted by the authors in English.