Geodesic
A Short Note on the Frobenius Norm of the Commutator
Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 934-939

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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 $\mathbb R^{n\times n}$. We first propose a new upper bound on basis of the Böttcher and Wenzel's inequality. Motivated by the method used, the inequality $\|\boldsymbol{XY}-\boldsymbol{YX}\|_F^2\le2\|\boldsymbol X\|_F^2\|\boldsymbol Y\|_F^2$ is finally improved into $$ \|\boldsymbol{XY}-\boldsymbol{YX}\|_F^2\le2\|\boldsymbol X\|_F^2\|\boldsymbol Y\|_F^2-2[\operatorname{tr}(\boldsymbol X^T\boldsymbol Y)]^2. $$ In addition, a further improvement is made.
Keywords: commutator, Böttcher and Wenzel's conjecture
Mots-clés : Frobenius norm, random matrix. 
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     author = {Yan-Dong Wu and Xu-Qing Liu},
     title = {A {Short} {Note} on the {Frobenius} {Norm} of the {Commutator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {934--939},
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     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a13/}
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Yan-Dong Wu; Xu-Qing Liu. A Short Note on the Frobenius Norm of the Commutator. Matematičeskie zametki, Tome 87 (2010) no. 6, pp. 934-939. http://geodesic.mathdoc.fr/item/MZM_2010_87_6_a13/