Conditions for the structural existence of an eigenvalue of a bipartite $(\min,\max,+)$-system¹

Jacob van der Woude & Subiono

Subfaculty of Technical Mathematics and Informatics
Faculty of Information Technology and Systems
Delft University of Technology
Mekelweg 4, 2628 CD Delft
the Netherlands

j.w.vanderwoude@math.tudelft.nl
subiono@math.tudelft.nl

Abstract:

In this talk we consider bipartite $(\min,\max,+)$-systems. We present conditions for the structural existence of an eigenvalue and corresponding eigenvector for such systems, where both the eigenvalue and eigenvector are supposed to be finite. The conditions are stated in terms of the system matrices that describe a bipartite $(\min,\max,+)$-system. Structural in the previous means that not so much the numerical values of the finite entries in the system matrices are important, rather than their locations within these matrices. The conditions presented in this talk can be seen as a version for bipartite $(\min,\max,+)$-systems of known conditions for the structural existence of an eigenvalue of a $(\max,+)$-system involving the (ir)reducibility of the associated system matrix.

Bibliography

1

M.V. Menon, Some spectral properties of an operator associated with a pair of non-negative matrices, Trans.Amer. Math. Soc., Vol 132, 1968, pp. 369-376.

2

G. J. Olsder, On structural properties of min-max systems, TWI report 93-95, Faculty of Technical Mathematics and Informmatices, 1993.