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In matrix $\textbf{A}=[a_{kj}]_{K\times K}$, each elemtent on the main diagonal is $a_{kk}=1$. Other elements is $0\leq a_{kj}\leq1$.

Besides, each non-diagonal elements satisfies $a_{kj}\geq a_{kl}a_{lj}, \forall l$.

$K$ is not so large, i.e., does not go to infinity.

Then does $\textbf{A}$ has some special properties? Can the property be describled by some inequalities or transformations?

What about the case $a_{kk}=0$?

Thanks a lot!

Severals-user45972
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  • It seems that $\textbf{A} \geq \frac{ \textbf{A}^n }{ K^{n-1} }$ holds. But it can not show something about $\textbf{A}$ directly. – Severals-user45972 Jul 17 '15 at 04:30
  • Not sure if this helps. If we define a relation $i\sim j$ whenever $a_{ij}\ne0$, then the relation $\sim$ is reflexive and transitive. So, by a relabelling of indices, you can at least rearrange $A$ into a block-upper triangular form, where the first column of each diagonal block is entrywise nonzero. – user1551 Jul 17 '15 at 13:00
  • Thanks. In fact, most elements in the matrix I am considering is non-zero. – Severals-user45972 Jul 18 '15 at 07:03

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