## A conjecture on a class of matrices.(English)Zbl 0616.15022

A non-negative $$m\times m$$ matrix $$A=\{a_{ij}\}$$ is called a Šidák matrix if it is irreducible and has at most one zero on its diagonal. If k(A) is the number of positive entries in A. Z. Šidák [ibid. 89, 28-30 (1964; Zbl 0143.263)] proved that $$k(A^ n)$$ is non-decreasing with n. If F(I) denotes the set of one-step consequent indices of a subset I of indices $$\{$$ 1,...,m$$\}$$ of A, then the author defines A to be connected if (i) it is irreducible; (ii) for any proper non-empty subset I of S, $$F(I)\cap F(I^ c)\neq \emptyset$$. He then shows that a Šidák matrix is a connected matrix, using a notion of equivalence of indices; and conjectures that $$k(A^ n)$$ is non-decreasing for any connected A. An unmentioned paper generalizing Šidák’s result on monotonicity of $$k(A^ n)$$ to a wider class of A than Šidák matrices by A. Vrba [ibid. 98, 298-299 (1973; Zbl 0263.05025)] is relevant. ”Vrba” matrices need not be irreducible; if irreducible and periodic, they will not be connected; if irreducible and aperiodic, they need not be connected, e.g. the $$3\times 3$$ matrix with $$a_{11}=a_{12}=a_{23}=a_{33}=0$$, other entries positive, has $$a_{13}a_{21}a_{32}>0$$.

### MSC:

 15B48 Positive matrices and their generalizations; cones of matrices 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)

### Citations:

Zbl 0143.263; Zbl 0263.05025
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