Hi gummy_ratz,

All you need to verify is that for any vector $\displaystyle x$ and any two positive definite matrices $\displaystyle A$ and $\displaystyle B$, then

$\displaystyle x^{T}(A+B)x = x^{T}Ax + x^{T}Bx \geq 0$.

First try and show this for a vector $\displaystyle (x_{1},x_{2})$ and $\displaystyle 2\times 2$ matrices

$\displaystyle \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)$ and $\displaystyle \left(\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right)$

Then you could try and do the general case.

Then do you see how this proves that $\displaystyle A+B$ is positive definite?