Notes for AKT-140124/0:23:30

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

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

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

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