# Estimating Spectral Radius of a Matrix Power (Special Case)

Consider the following equation: $$\Gamma = (I - P_nD^{-1}A)(I - P_{n-1}D^{-1}A)....(I - P_1D^{-1}A)$$ where:

$$A$$ is a diagonally dominant $$n$$ x $$n$$ matrix

$$D$$ is the diagonal matrix of A

$$I$$ is the identity matrix of size $$n$$ x $$n$$

$$P_i$$ is a partial identity matrix (not all diagonal elements are one)

and the rules on $$P_i$$ are:

$$P_n$$ + $$P_{n-1} + ... + P_1 = I \quad$$ and $$\quad P_n P_{n-1}...P_1 = 0$$

We also know that $$\rho(I - D^{-1}A) < 1$$.

Without $$P_i$$ matrices, it is possible to estimate the spectral radius of $$\Gamma$$. However, I cannot find a way to find $$\rho(\Gamma)$$ considering the $$P_i$$s. Numerical results show that $$\rho(\Gamma)$$ is less than one for any combination of $$P_i$$s. In fact, having more number of $$P_i$$ matrices will further decrease the value of $$\rho(\Gamma)$$.

I really appreciate any help to solve this problem. Thank you!