Let $\Sigma$ be a covariance matrix of vectors in $\mathbb{R}^n$. Given two orthogonal subspaces $B, B_\perp$ (such that $dim(B) + dim(B_\perp) = n$), is it possible to decompose $\Sigma = \Sigma_B + \Sigma_{B_\perp}$? I.e, a decomposition that separates between the contribution of each subspace to the total covariance?
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