**Reference:**

B. De Schutter and
B. De Moor,
"The QR decomposition and the singular value decomposition in the
symmetrized max-plus algebra revisited," *SIAM Review*, vol.
44, no. 3, pp. 417-454, 2002.

**Abstract:**

This paper is an updated and extended version of the paper "The QR
decomposition and the singular value decomposition in the symmetrized
max-plus algebra" (by B. De Schutter and B. De Moor, *SIAM Journal
on Matrix Analysis and Applications*, vol. 19, no. 2, pp. 378-406,
April 1998). The max-plus algebra, which has maximization and addition
as its basic operations, can be used to describe and analyze certain
classes of discrete-event systems, such as flexible manufacturing
systems, railway networks, and parallel processor systems. In contrast
to conventional algebra and conventional (linear) system theory, the
max-plus algebra and the max-plus-algebraic system theory for
discrete-event systems are at present far from fully developed and
many fundamental problems still have to be solved. Currently, much
research is going on to deal with these problems and to further extend
the max-plus algebra and to develop a complete max-plus-algebraic
system theory for discrete-event systems. In this paper we address one
of the remaining gaps in the max-plus algebra by considering matrix
decompositions in the symmetrized max-plus algebra. The symmetrized
max-plus algebra is an extension of the max-plus algebra obtained by
introducing a max-plus-algebraic analogue of the -operator. We show
that we can use well-known linear algebra algorithms to prove the
existence of max-plus-algebraic analogues of basic matrix
decomposition from linear algebra such as the QR decomposition, the
singular value decomposition, the Hessenberg decomposition, the LU
decomposition, and so on. These max-plus-algebraic matrix
decompositions could play an important role in the max-plus-algebraic
system theory for discrete-event systems.

Online version of the paper

Corresponding technical report: pdf file (422 KB)

@article{DeSDeM:02-012,

author={B. {D}e Schutter and B. {D}e Moor},

title={The {QR} decomposition and the singular value decomposition in the symmetrized max-plus algebra revisited},

journal={SIAM Review},

volume={44},

number={3},

pages={417--454},

year={2002},

doi={10.1137/S00361445024039}

}

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