Analysis Commons^™

Characterization And Properties Of $(R,S_\Sigma)$-Commutative Matrices, William F. Trench

William F. Trench

Let $R=P \diag(\gamma_{0}I_{m_{0}}, \gamma_{1}I_{m_{1}}, \dots, \gamma_{k-1}I_{m_{k-1}})P^{-1}\in\mathbb{C}^{m\times m}$ and $S_{\sigma}=Q\diag(\gamma_{\sigma(0)}I_{n_{0}},\gamma_{\sigma(1)}I_{n_{1}}, \dots,\gamma_{\sigma(k-1)}I_{n_{k-1}})Q^{-1}\in\mathbb{C}^{n\times n}$, where $m_{0}+m_{1}+\cdots +m_{k-1}=m$, $n_{0}+n_{1}+\cdots+n_{k-1}=n$, $\gamma_{0}$, $\gamma_{1}$, \dots, $\gamma_{k-1}$ are distinct complex numbers, and $\sigma :\mathbb{Z}_{k}\to\mathbb{Z}_{k}= \{0,1, \dots, k-1\}$. We say that $A\in\mathbb{C}^{m\times n}$ is $(R,S_{\sigma})$-commutative if $RA=AS_{\sigma}$. We characterize the class of $(R,S_{\sigma})$-commutative matrrices and extend results obtained previously for the case where $\gamma_{\ell}=e^{2\pi i\ell/k}$ and $\sigma(\ell)=\alpha\ell+\mu \pmod{k}$, $0 \le \ell \le k-1$, with $\alpha$, $\mu\in\mathbb{Z}_{k}$. Our results are independent of $\gamma_{0}$, $\gamma_{1}$, \dots, $\gamma_{k-1}$, so long as they are distinct; i.e., if $RA=AS_{\sigma}$ for some choice of $\gamma_{0}$, $\gamma_{1}$, \dots, $\gamma_{_{k-1}}$ (all distinct), then $RA=AS_{\sigma}$ for arbitrary of …