Mathematical Notes

, Volume 87, Issue 5, pp 903-907

First online:

A short note on the Frobenius norm of the commutator

  • Yan-Dong WuAffiliated withHuaiyin Institute of Technology
  • , Xu-Qing LiuAffiliated withHuaiyin Institute of Technology Email author 

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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 ‖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.

Key words

commutator Frobenius norm Böttcher and Wenzel’s conjecture random matrix