I am stuck on a problem in matrix algebra and I would be happy if someone could help me.
Given a square matrix with dimensions "p" given that $\textbf{x}$ $\sim$ N($\mu,\Sigma$) [multivariate normal], how can we show that: E[$\textbf{x}^{T} \textbf{A}\textbf{x}$] = Tr($\textbf{A}\Sigma$) + $\mu^{T} \textbf{A} \mu$ ?
$x^T A x = tr(x^T A x) = tr(x x^T A)$.
By linearity of expectation, $E[x^T A x] = E[tr(x x^T A)] = tr(E[x x^T A]) = tr(E[x x^T]A) = tr( (\Sigma + \mu \mu^T) A) = tr(\Sigma A) + tr(\mu \mu^T A) = tr(A \Sigma) + tr(\mu^T A \mu) = tr(A \Sigma) + \mu^T A \mu$
by the fact that trace is linear and invariant under cyclic permutations and trace of a scalar is itself.