View source for Notes for AKT-140124/0:23:30

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

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

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

From the above, we see that <math>-\frac12 <x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)>  + \frac12<y,\Lambda^{-1}y> = -\frac12<x,\Lambda x> + <x, y></math>