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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)
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