Remarks on the Bottcher-Wenzel Inequality

In 2005, B"ottcher and Wenzel raised the conjecture that if $X,Y$ are real square matrices, then $\mathrm{\mid }\mathrm{\mid }XY-YX\mathrm{\mid }{\mathrm{\mid }}^2\le 2\mathrm{\mid }\mathrm{\mid }X\mathrm{\mid }{\mathrm{\mid }}^2\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }{\mathrm{\mid }}^2$, where $\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }$ is the Frobenius norm. Various proofs of this conjecture were found in the last few years by several authors. We here give another proof. This proof is highly conceptual and requires minimal computation. We also briefly discuss related inequalities, in particular, the classical Chern-do Camo-Kobayashi inequality.