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.
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BibTeX
@article{DeSDeM:02-012,
author = {De Schutter, Bart and De Moor, Bart},
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}
}