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Operators preserving primitivity for matrix pairs. (English) Zbl 1215.15030

Olshevsky, Vadim (ed.) et al., Matrix methods. Theory, algorithms and applications. Dedicated to the memory of Gene Golub. Based on the 2nd international conference on matrix methods and operator equations, Moscow, Russia, July 23–27, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-283-601-4/hbk). 2-19 (2010).
From the introduction: Let \(\mathcal S\) be a semiring and \({\mathcal M}(\mathcal S)\) the set of matrices with entries from \(\mathcal S\). We investigate the structure of surjective additive transformations on the Cartesian product \({\mathcal M}^2({\mathcal S})={\mathcal M}({\mathcal S})\times {\mathcal M}({\mathcal S})\) preserving primitive matrix pairs. It turns out that for the characterization of these transformations we have to apply different and more involved techniques and ideas, such as primitive assignments, cycle matrices, etc.
Our paper is organized as follows: in Section 2 we collect some basic facts, definitions and notations, in Section 3 we characterize surjective additive transformations \(T : {\mathcal M}^2(\mathbb B)\to{\mathcal M}^2(\mathbb B)\) with \(\mathbb B\) a Boolean semiring preserving the set of primitive matrix pairs, in Section 4 we extend this result to matrices over arbitrary antinegative semiring without zero divisors.
For the entire collection see [Zbl 1202.15006].

MSC:

15A86 Linear preserver problems
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15B48 Positive matrices and their generalizations; cones of matrices
16Y60 Semirings
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