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I have a matrix $A$ of size $n \times n$ which consists of four blocks: $A = [B C; D E]$ where $B$ is an $k \times k$ matrix, $C$ is $k \times (n-k)$, $D$ is $(n-k) \times k$ and $E$ is $(n-k) \times (n-k)$ diagonal matrix.

I am wondering if there is a way to express the SVD of $A = U \Lambda V^{\top}$ (i.e. the projection matrices $U$, $V$ and the singular values) as function of the SVDs of $B$, $C$, $D$ and $E$?

(note again that $E$ is diagonal.)

(if it helps, all values in $A$ are positive.)

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