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Let be $\boldsymbol{S}$ an $N\times K$ real matrix having a block structure: \begin{equation} \boldsymbol{S} = \sum_{i=1}^{m} \boldsymbol{e}_i \otimes \boldsymbol{S}_i = \begin{bmatrix}\boldsymbol{S}_1 \\ \boldsymbol{S}_2 \\ \vdots \\ \boldsymbol{S}_m \end{bmatrix} \end{equation} where $\boldsymbol{e}_i$ is the $i$-th vector of the canonical basis, that is $[\boldsymbol{e}_i]_j=\delta_{i,j}$, and $\boldsymbol{S}_i$ a real $M\times K$ matrix $\forall i$. Of course, $N=m\cdot M$.

Is there any relationship between the singular values of $\boldsymbol{S}$ and those of $\boldsymbol{S}_i$?

P.S. For example, when $\boldsymbol{S}=\boldsymbol{A}\otimes \boldsymbol{B}$, there is a relationship between the eigenvalues of $\boldsymbol{A}$ and those of $\boldsymbol{B}$ (suppose that they are square matrices). In fact $\lambda_i(\boldsymbol{A})\lambda_j(\boldsymbol{B})$ are the eigenvalues of $\boldsymbol{S}$.

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Is the Cauchy interlace theorem of any help here? –  Dominique Aug 18 '13 at 17:10
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