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Background

Let $\mathcal{D}$ be the set of $S\times S$ diagonal matrices with elements in $[0,1)$. Let $\mathcal{G} = \{\Gamma \in \mathcal{D} : tr(\Gamma) = 1\}$, and let $B$ be an $S\times S$ matrix with $1$ in every entry.

Define the set of matrices $$ \mathcal{M} := \{ (\Gamma B - I)\Theta : \Gamma \in \mathcal{G}, \; \Theta \in \mathcal{D}\}. $$

It can be shown that :

  1. $\rho(M) \leq \hat{\theta} < 1$, $\forall M\in \mathcal{M}$, where $\hat{\theta}$ is the maximum element of the matrix $\Theta$ that defines $M$; and
  2. $\rho(\prod_{i=1}^n M_i) \leq \prod_{i=1}^n\hat{\theta}_i < 1$, $\forall M_i \in \mathcal{M}$, where $\hat{\theta}_i$ is the maximum element of the matrix $\Theta_i$ that defines $M_i$.

Now, define the following $3\times 3$ blocks matrices $$ H_1(M_1) = \left(\begin{array}{ccc} 0 & M_1& M_1 \\\\ 0 & I & 0 \\\\ 0 & 0 & I\\ \end{array}\right), \; H_2(M_2) = \left(\begin{array}{ccc} I & 0 & 0 \\\\ M_2 & 0 & M_2 \\\\ 0 & 0 & I\\ \end{array}\right),\; \mbox{and} \\\\ H_3(M_3) = \left(\begin{array}{ccc} I & 0 & 0 \\\\ 0 & I & 0 \\\\ M_3 & M_3 & 0\\ \end{array}\right), $$ and the set of $3\times 3$ block matrices $$ \mathcal{J} = \{H_3(M_3)H_2(M_2)H_1(M_1) : M_3, M_2, M_1 \in \mathcal{M} \}. $$

Note that $\rho(H_i) = 1$ and $\rho(H_iH_j) = 1$, $\forall i, j$.

Questions

Can it be shown that:

  1. $\rho(J) < 1$, $\forall J \in \mathcal{J}$ ? And, more generally
  2. $\rho(\prod_{i=1}^nJ_i)^{1/n} < 1, \forall n$, and $\forall J_i \in \mathcal{J}$ ?

If the above two inequalities are true for $3\times 3$ block matrices, can these inequalities be shown to be true for $K\times K$ block matrices, for $K \geq 4$?

Observations

  1. These inequalities can be proved for $K=2$ using the two inequalities given in the Background.
  2. Numerically it was observed that these two inequalities are true for several values of $K$ and $S$ and for a large number of randomly generated $M$ matrices.
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