The QR decomposition and the singular value decomposition in the symmetrized max-plus algebra

The max-plus algebra has maximization and addition as basic operations, and can be used to model a certain class of discrete event systems. In contrast to linear algebra and linear system theory many fundamental problems in the max-plus algebra and in max-plus-algebraic system theory-still need to be solved. In this paper we discuss max-plus-algebraic analogues of some basic matrix decompositions from linear algebra that play an important role in linear system theory. We use algorithms from linear algebra to prove the existence of max-plus-algebraic analogues of the QR decomposition and the singular value decomposition.