1. ## positive definite matrix *Urgent*

Show if K is positive definite then so is K^2

2. Originally Posted by chiph588@
Show if K is positive definite then so is K^2
The matrix $K$ is positive definite if for all real non-zeros vectors $x$:

$x^TKx>0$

$K$ is positive definite, so consider:

$x^TK^2x$

for any non-zero real vector $x$, then:

$x^TK^2x=x^TKKx=\frac{x^TKxx^TKx}{|x|^2}=\frac{(x^T Kx)(x^TKx)}{|x|^2}>0$

etc.

CB

3. why does $x^T KKx = \frac{x^T Kxx^T Kx}{|x|^2}$?

4. Originally Posted by chiph588@
why does $x^T KKx = \frac{x^T Kxx^T Kx}{|x|^2}$?
Because $x^Tx$ is a scalar equal to $|x|^2$

CB