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Sign patterns that allow eventual positivity. (English) Zbl 1190.15031

A real square matrix \(A\) is called eventually-positive if there exists a positive integer \(k_0\) such that for all \(k \geq k_0,~A^k\) is entrywise positive. A sign pattern matrix is a matrix having entries in \(\{+,-, 0 \}\). By \(sgn(A)\) we mean the sign pattern matrix having entries that are the signs of the corresponding entries in \(A\). For a sign pattern matrix \(\mathcal A\), the sign pattern class of \(\mathcal A\), denoted by \(\mathcal Q (\mathcal A)\) is the set of all \(A \in \mathbb{R}^{n \times n}\) such that \(sgn (A)=\mathcal A\). For a given property \(\mathcal P\) of a real matrix, we say that a sign pattern \(\mathcal A\) requires \(\mathcal P\) if every \(A \in \mathcal Q (\mathcal A)\) has property \(\mathcal P\). A sign pattern \(\mathcal A\) allows \(\mathcal P\) (or is potentially \(\mathcal P\)), if there exists \(A \in \mathcal Q (\mathcal A)\) that has property \(\mathcal P\).
Many properties have been studied in connection with characterizations of sign patterns that require or allow a particular property; see for instance M. Catral, D. D. Olesky and P. van den Driessche [Linear Algebra Appl. 430, No. 11–12, 3080–3094 (2009; Zbl 1165.15009)] and F. J. Hall and Z. Li [{L. Hogben} (ed.), Handbook of Linear Algebra. Discrete Mathematics and its Applications. Boca Raton, FL: Chapman & Hall/CRC (2007; Zbl 1122.15001)]. A sign pattern \(\mathcal A\) for which there exists \(A \in \mathcal Q (\mathcal A)\) such that \(A\) is eventually positive is called a potentially eventually positive (PEP) sign pattern matrix.
The authors investigate the question: what sign pattern matrices are PEP? First they present a sufficient condition that guarantees that a sign pattern is PEP. Then, they demonstrate by means of an example that a plausible necessary condition proposed by C. R. Johnson and P. Tarazaga [Positivity 8, No. 4, 327–338 (2004; Zbl 1078.15018)] for eventually positive matrices does not hold. Next, modifications to PEP sign patterns that yield additional PEP sign pattern matrices are studied. In the next section, it is shown that for \(n \geq 2\), an \(n \times n\) sign pattern matrix that has exactly one positive entry in each row and column is not PEP. As a corollary, it is shown that for \(n \geq 2\), the minimum number of \(+\) entries in an \(n \times n\) PEP sign pattern is \(n + 1\). Then, the authors, in a series of results, present classes of sign pattern matrices that are not PEP. In the final section, they classify all sign pattern matrices of orders \(2\) and \(3\) that either allow or do not allow eventual positivity.

MSC:

15B35 Sign pattern matrices
15B48 Positive matrices and their generalizations; cones of matrices
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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