Working with the generalized covariance formula for a vector $x$, I have the following:
$$E[(x-\mu)(x-\mu)^T)] = E(xx^T) – \mu E(x^T)$$
But the term $E(x^T)$ doesn’t make much sense to me. Does anyone have an idea why I’m getting this term with my matrix algebra?
The short answer is “yes”, $E(x^T) = E(x)^T=\mu^T$. Your full expression will be:
$E[(x−μ)(x−μ)^T)]=E(xx^T)−μE(x^T)-E(x)\mu^T+\mu\mu^T = E(xx^T)-\mu\mu^T$
The expectation operator doesn’t care about the shape of the vector or matrix it operates on.