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{ \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 }[/math].

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

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