Mathematical Notes

, Volume 87, Issue 5–6, pp 903–907 | Cite as

A short note on the Frobenius norm of the commutator

Article

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 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 

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References

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    A. Böttcher and D. Wenzel, “How big can the commutator of two matrices be and how big is it typically?,” Linear Algebra Appl. 403, 216–228 (2005).MATHCrossRefMathSciNetGoogle Scholar
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    A. Böttcher and D. Wenzel, “The Frobenius norm and the commutator,” Linear Algebra Appl. 429(8–9), 1864–1885 (2008).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Huaiyin Institute of TechnologyHuaiyinChina

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