Notes for AKT-140124/0:23:30

Let [math]\displaystyle{ \Lambda }[/math] be a symmetric, positive definite, non-singular square matrix. Then we have the following:

[math]\displaystyle{ \lt x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)\gt = \lt x,\Lambda x\gt - \lt x, y\gt -\lt \Lambda^{-1}y, \Lambda x\gt + \lt \Lambda^{-1}y,y\gt }[/math].

We have [math]\displaystyle{ \lt \Lambda^{-1}y, \Lambda x\gt = \lt x,y\gt }[/math] and [math]\displaystyle{ \lt \Lambda^{-1}y,y\gt = \lt y,\Lambda^{-1}y\gt }[/math] since [math]\displaystyle{ \Lambda }[/math] is symmetric.

From the above, we see that [math]\displaystyle{ -\frac12 \lt x - \Lambda^{-1} y, \Lambda(x - \Lambda^{-1}y)\gt + \frac12\lt y,\Lambda^{-1}y\gt = -\frac12\lt x,\Lambda x\gt + \lt x, y\gt }[/math]