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One-sided Max-linear Systems and Max-algebraic Subspaces

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Max-linear Systems: Theory and Algorithms

Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

Two approaches for solving one-sided max-linear systems are presented, combinatorial and algebraic. This fact highlights two fundamental aspects of max-algebra: its combinatorial nature and a specific way of dealing with the virtual non-existence of inverse matrices due to the lack of subtraction. In the latter case the significance of dual operators, matrix conjugation and extension of the ground set by an upper bound is demonstrated.

Since the solvability question is essentially deciding whether a vector is in a subspace (column space of a matrix), later in this chapter a theory of max-algebraic subspaces is presented. This includes the concepts of generators, independence and bases. A description of bases for general max-algebraic subspaces is given using extremals; it then follows that every finitely generated subspace has a unique scaled basis. Dimensional anomalies and unsolvable systems are also discussed.

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Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Birmingham, UK

    Peter Butkovič

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  1. Peter Butkovič
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Butkovič, P. (2010). One-sided Max-linear Systems and Max-algebraic Subspaces. In: Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84996-299-5_3

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