Prove this Matrix is Positive Definite Suppose 
u $ \neq 0 \in$ $\mathbb{C}^m$
$\alpha \in \mathbb{R}$
For what Values of $\alpha$ is $I + \alpha uu^{*}$ Positive Definite?
Progress so far :
$\forall x \neq 0 \in \mathbb{C}^m$ 
We Have $x^*Ix$ > 0
I cant determine the values of $\alpha$ which make the overall quantity Positive Definite however.

Hint:
$$
x^*\alpha uu^* x = \alpha (x^*u)(u^*x) = \alpha (x^*u)\overline{(x^*u)} = \alpha |x^*u|^2
$$
then use Cauchy-Schwarz; this will yield an $|x|^2$ to match $x^*Ix = |x|^2$.
