# Schur complement question

1. Apr 8, 2010

### math8

Let M be a real symmetric and positive definite matrix with blocks A, Bt, B and C.

$$M= [[A,B^{t}] ; [B,C]]$$

where $$A$$ is a $$p\times p$$ matrix;$$B$$ is $$q\times p$$; and $$C$$ is $$q\times q$$.

Let $$S=C-BA^{-1}B^{t}$$ be the Schur complement. We prove that S is symmetric positive definite.

I can prove that S is symmetric but I am having trouble proving that it is positive definite.
I know that for S a symmetric matrix, S positive definite is equivalent to say that all eigen values of S are positive.

I guess my question is how do we prove that all eigen values of S are positive?