Here $X,Y$ are vector of length $n$ and $A$ is a $n\times n$ matrix. Suppose the covariance matrix $D(X)$ is known?


This (linear transform) is typically listed as a property of covariance, but easy to show as well: $$\begin{align}\operatorname{cov}(AX)&=\mathbb E[AXX^TA^T]-\mathbb E[AX]\mathbb E[X^TA^T]\\&=A\mathbb E[XX^T]A^T-A\mathbb E[X]\mathbb E[X^T]A^T\\&=A(\mathbb E[XX^T]-\mathbb E[X]\mathbb E[X^T])A^T\\&=ADA^T\end{align}$$

  • 1
    $\begingroup$ I think in the general case, for complex-valued variables, those super-T should be super-H $\endgroup$
    – Luis Mendo
    May 11 '20 at 23:07

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