๐”– Bobbio Scriptorium
โœฆ   LIBER   โœฆ

A note on the characteristic equation in the max-plus algebra

โœ Scribed by Bart De Schutter; Bart De Moor

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
635 KB
Volume
261
Category
Article
ISSN
0024-3795

No coin nor oath required. For personal study only.

โœฆ Synopsis

We discuss the characteristic equation of a matrix in the max-plus algebra.

In their Linear Algebra Appl. paper [101:87-108 (1988)] Olsder and Roos have used a transformation between the max-plus algebra and linear algebra to show that the Cayley-Hamilton theorem also holds in the maw-plus algebra. We show that the derivation of Olsder and Roos is not entirely correct, and we give the correct formulas for the coefficients of this alternative version of the ma-algebraic characteristic equation. We also give a counterexample for a conjecture of Olsder in which he states necessary and sufficient for the coefficients of the max-algebraic characteristic equation.

๐Ÿ“œ SIMILAR VOLUMES

On the boolean minimal realization probl
On the boolean minimal realization problem in the max-plus algebra
โœ Bart De Schutter; Vincent Blondel; Remco de Vries; Bart De Moor ๐Ÿ“‚ Article ๐Ÿ“… 1998 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 127 KB

One of the open problems in the max-plus-algebraic system theory for discrete event systems is the minimal realization problem. In this paper we present some results in connection with the minimal realization problem in the max-plus algebra. First we characterize the minimal system order of a max-li

The max-plus algebra of the natural numb
The max-plus algebra of the natural numbers has no finite equational basis
โœ Luca Aceto; Zoltรกn ร‰sik; Anna Ingรณlfsdรณttir ๐Ÿ“‚ Article ๐Ÿ“… 2003 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 206 KB

This paper shows that the collection of identities which hold in the algebra N of the natural numbers with constant zero, and binary operations of sum and maximum is not รฟnitely based. Moreover, it is proven that, for every n, the equations in at most n variables that hold in N do not form an equati

On the power method in max algebra
On the power method in max algebra
โœ Ludwig Elsner; P.van den Driessche ๐Ÿ“‚ Article ๐Ÿ“… 1999 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 140 KB

The eigenvalue problem for an irreducible nonnegative matrix e ij in the max algebra system is e x kx, where e x i max j ij x j and k turns out to be the maximum circuit geometric mean, le. A power method algorithm is given to compute le and eigenvector x. The algorithm is developed by using results

An efficient algorithm for critical circ
An efficient algorithm for critical circuits and finite eigenvectors in the max-plus algebra
โœ Geert-Jan Olsder; Kees Roos; Robert-Jan van Egmond ๐Ÿ“‚ Article ๐Ÿ“… 1999 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 97 KB

We consider the eigenvalue problem in the max-plus algebra for matrices in fร€I Rg nร‚n but with eigenvectors in R n . The problem is relaxed to a linear optimization (LO) problem of which the dual problem is solved by ยฎnding a maximal average weight circuit in the graph of the matrix. The FloydยฑWarsh