Modeling of Low Rank Time Series

The paper’s Theorem 3 derives F(z) = [F0(z), F1(z)] with F0(z) = C2(zI−Γ1)^{-1}[I−B1(C1B1)^{-1}C1]B0Σ^{-1} and F1(z) = z C2(zI−Γ1)^{-1}B1(C1B1)^{-1}, and the equivalent form F1(z) = C2Γ1(zI−Γ1)^{-1}B1(C1B1)^{-1} + C2B1(C1B1)^{-1} (equations (34a–c)), under the SVD-based block-diagonalization of D and the partitioning of C and B; it also proves stability iff Γ1 is Schur (Corollary 1). These statements are established in the paper via a Schur-complement/inversion route building on (20)–(27) and Lemma 3 (the inversion lemma) . The candidate solution independently rederives the same formulas by direct elimination in state space using the projector P = I − B1(C1B1)^{-1}C1 and Γ1 = Γ0 − B1(C1B1)^{-1}C1Γ0, and arrives at the same stability criterion; this aligns with the paper’s setup for W(z) and partitions . Differences are methodological, not substantive.