I'm trying to bound the spectral radius of a matrix product. The first matrix is $$H = (I-M)^{-1}\Gamma,$$ where $\sum_jM_{ij}<1$, $M_{ij}\geq0$, and $\Gamma$ is a diagonal matrix with non-negative entries (when $\Gamma$ is a zero matrix then the question is trivial, so let’s focus on the case when $\Gamma$ has at least one non-zero entry).
Then the target matrix is $KH$. Matrix $K=P-Q$ is the difference of two right stochastic matrices (a right stochastic matrix has each row sums to 1 and all elements are non-negative), naturally |$K_{ij}| \leq 1$ and $\sum_{j}K_{ij} = 0$. In addition, $Q=\vec{1}\vec{q}'$ is a rank-1 matrix. That implies $|K_{ij}-K_{kj}|\leq1$.
The hypothesis is
$$\rho(KH) = \rho(HK) \leq \rho(H).$$
Alternatively, a bound provided by the norm of H would also be good. The corresponding conjecture is $$\rho(KH)(=\rho(HK))\leq||H||. $$
Does anybody have an idea of how this could be proved or whether this is right? Thanks in advance.
So far I've proved
$$\rho(HK) \leq 2 \|H\|_\infty$$
by the following reasoning: ($\|\cdot\|_\infty$ provides an upper bound of the spectral radius of a matrix, according to Gershgorin Disk Theorem.) \begin{align} \rho(HK) & \leq \|HK\|_\infty\\ &=\max_j\sum_i|HK_{ji}|\\ &=\max_j\sum_i|\Sigma_kH_{jk}K_{ki}| \\ &\leq \max_j\sum_i\Sigma_k|K_{ki}||H_{jk}|\\ &\leq \max_j2\sum_k|H_{jk}|\\ &=2\|H\|_\infty \end{align}
Actually any bound tighter than this would be great to see!
Do you think the result depends on the properties of the matrix $M$? I ask because the result is related to $H$ and not to $M$. In fact, the result you have already proven is in terms of $H$, not $M$. Or perhaps you think that it is only when $H$ satisfies these constraints (i.e., that the matrix $M$ associated with $H$ satisfies $\sum_j M_{ij}<1$ and $_{}\geq0$) that you can lower the $2$ to a $1$.
Now, if it is the case that one needs to use the structure of the underlying $M$ matrix, then perhaps using the expansion $(I-M)^{-1} = I + M + M^2 ...$ could help.