# error covariance of MMSE estimator relation to other error covariance estimators

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