How do I show the covariance matrix of a multivariate normal random vector is positive definite?

Question

The question is as follows:

Suppose the $n$-dimensional random vector $\textbf{Z}$ has mean vector $\mu$ and variance-covariance $V$. By considering $Var(x^{T}\textbf{Z})$ for $x \in \mathbb{R}^n$, show that $V$ is positive semi-definite.

So, $Var(x^{T}\textbf{Z})$ = $x^TVar(\textbf{Z})x = x^{T}Vx.$ But i'm unaware to why this tells me $V$ is positive semi-definite. I feel like i'm painfully close...

Sam

Answer 1

Let $\text{Var}(Z)$ be $n\times n$. The definition of semi-positive definiteness is that $x^T\text{Var}(Z)x \ge 0$ for all $n$-dimensional vectors. Choose such a $x$, then as you pointed out $$ x^T\text{Var}(Z)x = \text{Var}(xZ) \ge 0 $$ by definition of the variance. This proves the statement.