13
$\begingroup$

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.

$\endgroup$
14
$\begingroup$

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)

$\endgroup$
  • $\begingroup$ Why is $Tr(X^T AX) = X^T AX$? $\endgroup$ – Aqqqq May 26 '18 at 14:25
  • 1
    $\begingroup$ Because $X^TAX$ is a number $\endgroup$ – kjetil b halvorsen May 26 '18 at 15:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.