Proof of differetial entropy of the Multivartiate Gaussian

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How do you get from the left hand side to the right hand side of this equation, this is converting the multivariate gaussian in to differential entropy.

$\mathbb{E}[(x-\mu)^T\Sigma^{-1}(x-\mu)]=\mathbb{E}[tr(x-\mu)^T\Sigma^{-1}(x-\mu)]$

How can we just put the trace in and remain equivalent?

Later on they pull the trace out of the expectation, does this mean that the trace is a linear operator and why?