Show if K is positive definite then so is K^2
The matrix $\displaystyle K$ is positive definite if for all real non-zeros vectors $\displaystyle x$:
$\displaystyle x^TKx>0$
$\displaystyle K$ is positive definite, so consider:
$\displaystyle x^TK^2x$
for any non-zero real vector $\displaystyle x$, then:
$\displaystyle x^TK^2x=x^TKKx=\frac{x^TKxx^TKx}{|x|^2}=\frac{(x^T Kx)(x^TKx)}{|x|^2}>0$
etc.
CB