Characterization and properties of (R,Sσ)-commutative matrices

Let and , where m0+m1+⋯+mk-1=m, n0+n1+⋯+nk-1=n, γ0, γ1, …, γk-1 are distinct complex numbers, and σ:ZktoZk={0,1,…,k-1}. We say that A∈Cm×n is (R,Sσ)-commutative if RA=ASσ. We characterize the class of (R,Sσ)-commutative matrices and extend results obtained previously for the case where γℓ=e2πiℓ/k and σ(ℓ)=αℓ+μ(modk), 0⩽ℓ⩽k-1, with α, μ∈Zk. Our results are independent of γ0, γ1, …, γk-1, so long as they are distinct; i.e., if RA=ASσ for some choice of γ0, γ1,…,γk-1 (all distinct), then RA=ASσ for arbitrary of γ0, γ1, …, γk-1.