Here are some formula for matrix inversion.
Lemma 1: For a block matrix , , where is basically the Schur's complement of block .
Proof: Let , gives us
From the four equations, we have
And similarly from , we have
Together, they show the first inequality. Also note that and thus . Substituting into above shows the second equality. .
Lemma 2 (Matrix Inversion Formula/Woodbury Matrix Identity):
Proof: From the previous proof, we have . We get the identity immediately as we flip the sign of .