Characterization and properties of matrices with generalized symmetry or skew symmetry

Let be a nontrivial involution; i.e., R=R−1≠±I. We say that is R-symmetric (R-skew symmetric) if RAR=A (RAR=−A). There are positive integers r and s with r+s=n and matrices and such that P*P=Ir, Q*Q=Is, RP=P, and RQ=−Q. We give an explicit representation of an arbitrary R-symmetric matrix A in terms of P and Q, and show that solving Az=w and the eigenvalue problem for A reduce to the corresponding problems for matrices and . We also express A−1 in terms of APP−1 and AQQ−1. Under the additional assumption that R*=R, we show that Moore–Penrose inversion and singular value decomposition of A reduce to the corresponding problems for APP and AQQ. We give similar results for R-skew symmetric matrices. These results are known for the case where R=J=(δi,n−j+1)i,j=1n; however, our proofs are simpler even in this case. We say that is R-conjugate if where and R=R−1≠±I. In this case is R-symmetric and is R-skew symmetric, so our results provide explicit representations for R-conjugate matrices in terms of P and Q, which are now in and respectively. We show that solving Az=w, inverting A, and the eigenvalue problem for A reduce to the corresponding problems for a related matrix . If RT=R this is also true for Moore–Penrose inversion and singular value decomposition of A.