Let's say we have a positive-definite symmetric matrix $M$ with individual elements sample from a distribution containing an expected value of $0$.
$$M = Q \Lambda Q^{T}$$ $$E \left[ M\right] = 0 = E \left[Q \Lambda Q^{T} \right]$$
Now since we know that $M$ is symmetric, it is positive definite therefore it's eigenvalues are all greater than zero, therefore, we can say
$$0 \leq E\left[\Lambda\right]$$
How do we show that $ E \left[Q \right] = 0$. Is it as simple as stating $$E \left[ M\right]=E \left[Q \Lambda Q^{T} \right] = E \left[Q \right] * E \left[\Lambda \right]* E \left[Q^T \right]$$
And since the $\Lambda$ expectation is greater than 0, the only way this expression is true is $E \left[Q \right]$ is 0.
Is this a valid statement? Do we have to write the covariances? And if so how?