Suppose $x$ is a $n\times 1$ vector of random variables each of which has a finite second moment. Let $$ \mu=E(x),\quad V=E[(x-\mu)(x-\mu)']. $$ Then, $V$ is positive semidefinite. I'm wondering whether the converse is true:
if $V$, dimension $n\times n$, is positive semidefinite, must there exist some $n\times 1$ random vector for which $E(xx')=V$?
That is, I'm wondering if being the covariance matrix of a real random vector imposes any other property other than positive semidefiniteness.
Let $v_1, \ldots, v_n$ be some mutually orthogonal vectors in $\mathbb{R}^n$ such that $\lVert v_k \rVert^2 = n\lambda_k \geq 0$ then what is the covariance matrix of the set $\{\pm v_k \mid k=1, \ldots, n\}$ of $2n$ elements? What are its eigenvalues and eigenvectors?