If $X$ is a $p \times 1$ gaussian random vector with such that $X \sim \mathcal{N}(0,\Sigma)$. What is the expected value of the square of the euclidean norm i.e $E[\|AX\|_2^2]$? Here $A$ is a $n \times p$ matrix.
[Math] Expected value of square of Euclidean norm of a gaussian random vector
expectationnormal distribution
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Best Answer
Setting $Y=AX$ we have $Y\sim\mathcal N(0,A\Sigma A^T)$ and:
$$\mathsf E\|AX\|_2^2=\operatorname{\mathsf E} Y^TY = \sum_{i=1}^n \operatorname{\mathsf E} Y_i^2 = \sum_{i=1}^n \operatorname{\mathsf{Var}}Y_i=\operatorname{\mathsf{tr}}(A\Sigma A^T)$$