Polar Decomposition of a Semiseparable Matrix in Pontryagin Spaces

Given a nonsingular signature $J_2$ and a causal left invertible operator $G$ by a minimal, u.e.s realization, we state necessary and sufficient conditions in state-space terms under which the indefinite spectral factorization problem $G{J}_2{G}^{\ast }=G_o{J}_x{G}_o^{\ast }$ has solution for a left-outer $G_o$ and another invertible signature $J_x$. We also state a single-pass numerically stable algorithm that finds $J_x$ and a minimal, u.e.s realization for $G_o$.