I'm trying to prove the following:

let $ \Lambda_{e}$ be the error covariance of an estimator $\,\hat{\theta}(y)$ of $\,\theta$ based on $\,y$. I want to show that the error covariance of MMSE estimator satisfies: $\Lambda_{MMSE}\le \Lambda_{e}$ i.e. the matrix $\Lambda_{MMSE}- \Lambda_{e}$ is PSD

what I thought of is:

$\Lambda_{MMSE}- \Lambda_{e}$ is symmetric thus if ill show that $v^{T}(\Lambda_{MMSE}- \Lambda_{e})v \le 0 \,\,\,\forall v$

now $v^{T}(\Lambda_{MMSE}- \Lambda_{e})v \le trace(\Lambda_{MMSE}- \Lambda_{e})\cdot\|v\|^{2}$ since $\|v\|^{2}\ge0\, \forall v$

$trace(\Lambda_{MMSE}- \Lambda_{e})\le0$

and now im stuck

would appreciate some help


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