# SVD of “almost” block diagonal matrix

It is possible to show that SVD of block diagonal matrix is equivalent to independent SVDs of each block. I am wondering if there is something interesting to say on the case where the matrix is composed of 2 blocks of size $$N$$, with $$k$$ rows and $$k$$ columns of size $$2N+k$$ (overlapping in a $$k\times k$$ corner) added on. I am particularly interested in the influence of the “shared” rows on the reconstruction of the entire matrix when using truncated SVD.

An example:

$$\begin{pmatrix} \begin{matrix} X_1 \end{matrix} & 0 & \vdots \\ \ 0& X_2 & \vec{u} \\ \cdots & \vec{v} &\vdots \end{pmatrix}$$

Here, $$X_1$$ and $$X_2$$ are the two diagonal blocks; each block is $$N \times N$$. I concatenate to them a single row $$\vec{v}$$ with $$2N+1$$ entries and a single column $$\vec{u}$$ with $$2N+1$$ entries, and aim to understand how the SVD of the complete matrix relates to the SVD of the block diagonal sub-matrix.

Thanks

• It's a bit difficult for me to understand what you mean by "$N+k$ dense rows of size $2N$". Could you explain a bit more or give an example?
– Max
Dec 18, 2020 at 2:01
• I added an example, I hope it's easier to understand now. Dec 18, 2020 at 22:42
• Yes, I think so. I edited a bit more, check if you agree. Now I think I understand what you are asking. (Still don't know if there is any useful relation of the kind you are asking for, though. Seems challenging).
– Max
Dec 18, 2020 at 22:56
• I believe that the matrices you're describing are called block arrowhead matrices. Dec 19, 2020 at 17:27
• Cauchy's interlacing theorem for singular values gives inequalities. The discussion here explains why the interlacing inequality holds for singular values. Dec 21, 2020 at 7:35