# Decomposition of the covariance matrix over two orthogonal subspaces

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?

Thanks

• What kind of decomposition would you expect? $\Sigma$ is not uniquely determined by its restriction to $B$ and $B^\perp$. How would you distinguish eg. $\begin{bmatrix}2&1\\1&2\end{bmatrix}$ and $\begin{bmatrix}2&0\\0&2\end{bmatrix}$ on $B=\langle e_1 \rangle$?
– md5
Commented Apr 18, 2020 at 20:29
• I don't have anything concrete in mind, but was wondering if any such decomposition is possible. Could you elaborate on "not uniquely determined by its restriction to $B$ and $B_\perp$"? What if we limited ourselves to the diagonal of the covariance matrix (the variances)? Commented Apr 18, 2020 at 20:42