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!


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