Matrix Identity

A useful matrix identity: \[ (P^{-1}+B^TR^{-1}B)^{-1}B^TR^{-1} = PB^T(BPB^T+R)^{-1} \] It can be proved with \[ \begin{aligned} (P^{-1}+B^TR^{-1}B)^{-1}B^TR^{-1} &= PB^T(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= (P^{-1}+B^TR^{-1}B)PB^T(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= (P^{-1}PB^T+B^TR^{-1}BPB^T)(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= (B^T+B^TR^{-1}BPB^T)(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= (B^TR^{-1}R+B^TR^{-1}BPB^T)(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= B^TR^{-1}(R+BPB^T)(BPB^T+R)^{-1} \\ \iff B^TR^{-1} &= B^TR^{-1} \\ \end{aligned} \] Its reduced form: \[ (I_N+AB)^{-1}A = A(I_M+BA)^{-1} \] It can be proved with \[ \begin{aligned} (I_N+AB)^{-1}A &= A(I_M+BA)^{-1} \\ \iff A &= (I_N + AB)A(I_M + BA)^{-1} \\ \iff A &= (A + ABA)(I_M + BA)^{-1} \\ \iff A &= A(I_M + BA)(I_M + BA)^{-1} \\ \iff A &= A \end{aligned} \]