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 .