Characterize those matrices $ X $ (real symmetric), $ Y$ (real positive definite), $ R$ (real diagonal) and $ F $ (real diagonal) such that
$ XRY + YRX = 0$ , (1)
$ YRY - XRX = F$ . (2)
i.e, can we give some explicit expressions (or give some properties) for $ X$ , $ Y$ , $ R$ , and $ F$ ?
From (1), I can only derive that
$RX = MJM^{-1}$ and $ Y = M^{-T}M^{-1}$ , where $ J$ is skew symmetric and $ M$ is full rank.