For the case we have an SBM model with two clusters of equal size (e.g. each of size n/2), we know that the following matrix: $$ \begin{bmatrix} J_{\frac{n}{2}} \cdot p & J_{\frac{n}{2}} \cdot q \newline J_{\frac{n}{2}} \cdot q & J_{\frac{n}{2}} \cdot p \end{bmatrix} $$ is of rank 2 and its eigenvalues and eigenvectors are $\frac{n}{2} \cdot (p+q)$, $\frac{n}{2} \cdot (p-q)$ and $\vec{1}$, $\begin{bmatrix} \frac{1}{\sqrt{n}}\\ -\frac{1}{\sqrt{n}} \end{bmatrix}$, respectively.

Is there any closed form solution for the eigenvalues and eigenvectors for the case where the clusters are not balanced, e.g. we have a matrix like this: $$ \begin{bmatrix} p & p & p & q & q \newline p & p & p & q & q \newline p & p & p & q & q \newline q & q & q & p & p \newline q & q & q & p & p \newline \end{bmatrix} $$ where this matrix is still of rank 2?

  • $\begingroup$ The linked duplicate gives the answer, but note that it uses $p, q$ for the dimensions of the blocks and not for the entries. $\endgroup$ Sep 28, 2022 at 0:12