Let $L$ and $\Psi$ be $p\times p$ matrices where $\Psi=diag(\psi_1,\dots,\psi_p)$. If $$F_i^M=(L^T\Psi^{-1}L)^{-1}L^T\Psi^{-1}(X_i-\mu)$$
$$F_i^R=L^T(LL^T+\Psi^{-1})(X_i-\mu)$$ where $X_i$ and $\mu$ are $p\times 1$ vectors. Verify that $$F_i^M=(L^T\Psi^{-1}L)^{-1}(I+L^T\Psi^{-1}L)F_i^R=(I+(L^T\Psi^{-1}L)^{-1})F_i^R$$
The first equality was easy to see because I just used a result that says $$L^T(LL^T+\Psi)^{-1}=(I+L^T\Psi^{-1}L)^{-1}L^T\Psi^{-1}$$ Then $$(L^T\Psi^{-1}L)^{-1}(I+L^T\Psi^{-1}L)F_i^R=(L^T\Psi^{-1}L)^{-1}(I+L^T\Psi^{-1}L)(I+L^T\Psi^{-1}L)^{-1}L^T\Psi^{-1}(X_i-\mu)$$ $$=(L^T\Psi^{-1}L)^{-1}L^T\Psi^{-1}(X_i-\mu)=F_i^M$$
I don't know how to check the last equality, I tried to use that $$(I+(L^T\Psi^{-1}L)^{-1})=I^{-1}(I+(L^T\Psi^{-1}L))(L^T\Psi^{-1}L)^{-1}$$ and substitute $F_i^R$ but didn't worked.