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Let $x\in \mathbb{R}^{d_x}, y\in \mathbb{R}^{d_y}$ be two (zero-mean) random vectors with joint probability distribution $\mathcal{D}$. Denote the auto- and cross-covariance matrices with $\Sigma_{x}:=\mathbb{E}[x x^\top]$, $\Sigma_{y}:=\mathbb{E}[y y^\top]$, $\Sigma_{x,y}:=\mathbb{E}[x y^\top]$, and $\Sigma_{y,x}:=\mathbb{E}[ yx^\top]$. Assume $\Sigma_x = \mathrm{I}$.

Under what conditions on the joint distribution $\mathcal{D}$ do we have that $\mathop{Trace}(\Sigma_{y,x}\Sigma_{x,y}) = \mathop{Trace}(\Sigma_y)$?

For example, this is the case if $y$ is a (deterministic) linear function of $x$, say $y=Mx$ for some real matrix $M\in \mathbb{R}^{d_y \times d_x}$. But is it necessary?

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  • $\begingroup$ If $\mathbb{E}(y \mid x) = M x$ then this still holds. $\endgroup$ – WimC Jul 20 '18 at 17:08

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