The answer is quite simple.
The correlation matrix is defined thus:
Let $X = [x_1, x_2, ..., x_n]$ be the $m\times n$ data matrix: $m$ observations, $n$ variables.
Define $X_b= [\frac{(x_1-\mu_1 e)}{s_1}, \frac{(x_2-\mu_2 e)}{s_2}, \frac{(x_3-\mu_3 e)}{s_3}, ...]$ as the matrix of normalized data, with $\mu_1$ being mean for the variable 1, $\mu_2$ the mean for variable 2, etc., and $s_1$ the standard deviation of variable 1, etc., and $e$ is a vector of all 1s.
The correlation matrix is then
$$C=X_b' X_b$$
divided by $m-1$.
A matrix $A$ is positive semi-definite if there is no vector $z$ such that $z' A z <0$.
Suppose $C$ is not positive definite. Then there exists a vector w such that $w' C w<0$.
However $(w' C w)=(w' X_b' X_b w)=(X_b w)'(X_b w) = {z_1^2+z_2^2...}$, where $z=X_b w$, and thus $w' C w$ is a sum of squares and therefore cannot be less than zero.
So not only the correlation matrix but any matrix $U$ which can be written in the form $V' V$ is positive semi-definite.