The Rao-Mitra-Bhimasankaram relation is strongly antisymmetric

Abstract

For $n\times m$ complex matrices A and B, the Rao-Mitra-Bhimasankaram (RMB) relation $\stackrel{\mathrm{RMB}}{\le }$, defined by $A\stackrel{\mathrm{RMB}}{\le }B$ if $A{A}^{⁎}A=A{B}^{⁎}A$, is reflexive and antisymmetric, but not transitive. This paper shows that, despite the lack of transitivity, $\stackrel{\mathrm{RMB}}{\le }$ is strongly antisymmetric in the sense that, for all integers $n\ge 2$, $A_1\stackrel{\mathrm{RMB}}{\le }\cdots \stackrel{\mathrm{RMB}}{\le }{A}_n\stackrel{\mathrm{RMB}}{\le }{A}_1$ implies $A_1=\cdots =A_n$. The proof of this result is based on a novel proof that $\stackrel{\mathrm{RMB}}{\le }$ is antisymmetric.