Characterization and properties of matrices with k-involutory symmetries

We say that a matrix R∈Cn×n is k-involutory if its minimal polynomial is xk-1 for some k⩾2, so Rk-1=R-1 and the eigenvalues of R are 1, ζ,ζ2,…,ζk-1, where ζ=e2πi/k. Let μ∈{0,1,…,k-1}. If R∈Cm×m, A∈Cm×n,S∈Cn×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS-1=ζμA. If R,A∈Cn×n, we say that A is (R,μ)-symmetric if RAR-1=ζμA. We show that an (R,S,μ)-symmetric matrix A can be represented in terms of matrices Fs∈Ccs+μ×ds,0⩽s⩽k-1, where cs and ds are, respectively, the dimensions of the ζs-eigenspaces of R and S and + denotes addition modulo k. The system Az=w can be solved by solving k independent systems with the matrices F0,F1,…,Fk-1. If A is invertible then A-1 is can be expressed in terms of . We do not assume in general that R and S are unitary; however, if they are then the Moore–Penrose inverse A† of A can be written in terms of , and a singular value decomposition of A can be written simply in terms of singular value decompositions of F0,F1,…,Fk-1. If A is (R,0)-symmetric then solving the eigenvalue problem for A reduces to solving the eigenvalue problems for F0,F1,…,Fk-1. We also solve the eigenvalue problem for the more complicated case where A is (R,μ)-symmetric with μ∈{1,…,k-1}.