Singular value decomposition of a matrix $\mathbf{A}\in\mathbb{C}^{r\times r}$ can be written as $$\mathbf{A}=\sum_{i=1}^{r}\sigma_i\mathbf{u}_i\mathbf{v}_i^H$$
However, I have a matrix $\mathbf{B}\in\mathbb{C}^{r\times r}$ which is known to be given by $$\mathbf{B}=\sum_{i=1}^{k}s_i\mathbf{p}_i\mathbf{q}_i^H$$ where $k > r$.
I could not find any result on how the $r$ singular values of $\mathbf{B}$ relate to $\{s_i\}$. The result would have been simple for $k\le r$ but the $k>r$ case seems to evade my research. I would appreciate any directions to any resources which might be useful!