A Short Note on the Frobenius Norm of the Commutator

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.