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**math8** Let M be a real symmetric and positive definite matrix with blocks A, Bt, B and C.

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

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

Let $\displaystyle S=C-BA^{-1}B^{t}$ 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?