Notes for AKT-140124/0:23:30

Let $\Lambda$ be a symmetric, positive definite, non-singular square matrix. Then we have the following:

$<x-\Lambda ^{-1}y,\Lambda (x-\Lambda ^{-1}y)>=<x,\Lambda x>-<x,y>-<\Lambda ^{-1}y,\Lambda x>+<\Lambda ^{-1}y,y>$.

We have $<\Lambda ^{-1}y,\Lambda x>=<x,y>$ and $<\Lambda ^{-1}y,y>=<y,\Lambda ^{-1}y>$ since $\Lambda$ is symmetric.

From the above, we see that $-{\frac {1}{2}}<x-\Lambda ^{-1}y,\Lambda (x-\Lambda ^{-1}y)>+{\frac {1}{2}}<y,\Lambda ^{-1}y>=-{\frac {1}{2}}<x,\Lambda x>+<x,y>$