# low rank decomposition of composite PSD matrix

The matrix $M=AVA^T - BCB^T +D$ is known to be positive semidefinite (PSD), where $V, C, D$ are each diagonal matrices with positive values, and $V, C$ has small size when compared to the size of $M$. $A, B$ has different (but small) number of columns, and they are not necessarily orthogonal matrices. $D$ has the same size as $M$.

I need an 'economic' decomposition of $M$ in the form of $M=QPQ^T + \Delta$, where $\Delta$ is diagonal and $P$ is positive semidefinite, sparse, and ideally has a small size. So the question is that how could I exploit that $M$ is built from small rank and sparse elements. The answer should contain formulae for $Q, P, \Delta$ with the $A,V,B,C,D$ matrices ($\Delta$ can be zero). A $P$ using $V,-C,D$ in diagonal blocks comes close, but because of $-C$ it is not PSD (but this does not exploit that $M$ is known to be PSD).

One possible solution would find the eigendecomposition or the Cholesky of $M$, but I am happy with anything until $P$ and $\Delta$ has the desired properties.

Thank you!

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