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Abstract
We study the transients of matrices in maxplus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by max{T_{1},T_{2}}, where T_{1} is the weak CSR threshold and T_{2} is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for T_{1} and T_{2}, naturally leading to the question which matrices, if any, attain these bounds. In the present paper, we characterize the matrices attaining two particular bounds on T_{1}, which are generalizations of the bounds of Wielandt and Dulmage–Mendelsohn on the indices of nonweighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the nonweighted case.
Original language  English 

Journal  Linear and Multilinear Algebra 
Early online date  3 Feb 2021 
DOIs  
Publication status  Epub ahead of print  3 Feb 2021 
Bibliographical note
FundingThis work was partially supported by Agence Nationale de la Recherche (ANR) Perturbations [grant number ANR10BLAN0106]. The work of S. Sergeev was also supported by Engineering and Physical Sciences Research Council (EPSRC) [grant number EP/P019676/1].
Keywords
 15A18
 15A23
 90B35
 Maxplus
 digraphs
 matrix powers
 periodicity
 transient
ASJC Scopus subject areas
 Algebra and Number Theory
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Dive into the research topics of 'On the tightness of bounds for transients of weak CSR expansions and periodicity transients of critical rows and columns of tropical matrix powers'. Together they form a unique fingerprint.Projects
 1 Finished

Tropical Optimisation
Engineering & Physical Science Research Council
1/04/17 → 31/08/19
Project: Research Councils