# Find the analytical form for the spectral radius of a special sparse matrix, or its order approaching 1

Let $$\hat{\bf H}$$ be a $$p\hat{N}\times p \hat{N}$$ sparse matrix consisting of $$p\times p$$ blocks, where each block is of size $$\hat{N}\times\hat{N}$$. The values in $$\hat{\bf H}$$ is illustrated below (empty places are zero):

Asking for help to find the analytic form of the spectral radius of this matrix. If this is hard, then finding the order of the spectral radius approaching 1 is also good for us. I did a simple numerical experiment as the following,

If we fix $$p = 5$$ and let $$\hat{N}$$ go from 5 to 100, we have

If we fix $$\hat{N} = 10$$ and let $$p$$ go from 5 to 100, we have

• What make you think that such a form exist ? It is very exceptional that such formulas exist... – Jean Marie Becker Dec 8 '18 at 6:11