Consider a stochastic vector $v$ ($v_i \geq 0$ and $\sum_i v_i = 1$), a strictly substochastic matrix $A$ ($a_{ij} \in [0,1]$, $\sum_j a_{ij} < 1$), and a substochastic diagonal matrix $D$ ($D_{ij} = 0$ if $i \neq j$ and $D_{ii} \in [0,1]$) . Let $B = (I-DA^T)^{-1}$ and let $$ \Omega(v,D) \equiv (diag(Bv))^{-1} B [ diag(v) D + diag(Bv) (I-D) ] B^T (I - diag(A \iota)), $$ where $\iota$ is the vector of all ones.
Numerical simulations reveal that the spectral radius of $\Omega(v,D)$ is always weakly lower than one, and achieves the maximum of one when $v$ is at a corner ($v_i = 1$ for some $i$) and $ D = I$. I would like to prove this numerical result, but don't know how to proceed.
(It may be that I need to impose conditions so that $b_{ij} > 0$ for all $i,j$, thus guaranteeing that $diag(Bv))^{-1}$ exists.)
I can see that the spectral radius of $\Omega(v,I)$ is less than or equal to 1. This follows because $$ \Omega(v,I) = diag(Bv))^{-1} B diag(v) (I-A)^{-1} (I - diag(A \iota)).$$ Both $diag(Bv))^{-1} B diag(v)$ and $(I-A)^{-1} (I - diag(A \iota))$ are stochastic matrices, and hence the spectral radius of $\Omega(v,I)$ is less than or equal to 1. Letting $w(k)$ be the vector with k-th element equal to 1 and the rest zeros, we can also see that $\Omega(w(k),I)$ is a matrix with all rows equal to the k-th row of matrix $(I-A)^{-1} (I - diag(A \iota))$. The spectral radius is then equal to the sum of row k of matrix $(I-A)^{-1} (I - diag(A \iota)),$ which is one.
I can prove this for the special case in which $A$ is diagonal. In that case $B$ is also diagonal and
$$ \Omega(v,D) = [ D + B (I-D) ] B (I - A). $$
Each element of this diagonal matrix is equal to
$$ \left( d_i + \frac{1-d_i}{1-d_i a_{ii}} \right) \frac{1-a_{ii}}{1-d_i a_{ii}},$$ which is lower than one except at $d = 1,$ in which case it is equal to one.
Numerical simulations also suggest that (1) for any $D$ and $v$ there is a $k$ such that $\rho(\Omega(w(k),D)) \geq \rho(\Omega(v,D))$, and that (2) for any $k$ we have $\rho(\Omega(w(k),I)) = 1 \geq \rho(\Omega(w(k),D))$. Thus, we can proceed sequentially, first proving (1) and then (2) to get our overall result.
Any clues on how to proceed would be much appreciated.
