I know from standard theory that the bias-variance decomposition for Mean Squared Error is (for an estimator $\hat{\mu}$ of $\mu$):
$$ E\left[\left(\hat{\mu}-\mu\right)^2\right] = Var(\hat{\mu}) + \left(E\left[\mu\right]-\mu\right)^2 = Var(\hat{\mu}) +Bias(\hat{\mu}, \mu)^2 $$
However, if today we are talking about vectors, where $\boldsymbol{\hat{\mu}}$ is an estimator of $\boldsymbol{\mu}$, both of which are $n\times 1$ vectors, I was wondering if there is a corresponding nice decomposition as in the scalar case above for:
$$ E\left[||\boldsymbol{\hat{\mu}}-\boldsymbol{\mu}||^2\right] $$ ?
Thanks!