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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.

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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, 2018 at 14:25
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    $\begingroup$ Because $X^TAX$ is a number $\endgroup$ May 26, 2018 at 15:07
  • $\begingroup$ @kjetilbhalvorsen: Thanks, I have added your proof here. $\endgroup$
    – Joram Soch
    Nov 3, 2021 at 16:32

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