For random variables $X \in \mathbb{R}^h$, and a positive semi-definite matrix $A$: Is there a simplified expression for the expected value, $\mathop {\mathbb E}[Tr(X^TAX)]$ and variance, $Var[Tr(X^TAX)]$? Please note that $A$ is not a random variable.


Since $X^TAX$ is a scalar, $$\text{Tr}(X^TAX)= X^TAX = \text{Tr}(AXX^T)$$ so that $$\text{E}(X^TAX) = \text{E}(\text{Tr}(AXX^T)) = \text{Tr}(\text{E} (A XX^T)) = \text{Tr}(A\text{E}(XX^T))$$.

Here we have used that the trace of a product are invariant under cyclical permutations of the factors, and that the trace is a linear operator, so commutes with expectation. The variance is a much more involved computation, which also need some higher moments of $X$. That calculation can be found in Seber: "Linear Regression Analysis" (Wiley)

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  • $\begingroup$ Why is $Tr(X^T AX) = X^T AX$? $\endgroup$ – Aqqqq May 26 '18 at 14:25
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    $\begingroup$ Because $X^TAX$ is a number $\endgroup$ – kjetil b halvorsen May 26 '18 at 15:07

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