Max-algebraic attraction cones of nonnegative irreducible matrices

✍ Scribed by Sergei˘ Sergeev

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 406 KB
• Volume: 435
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2011.02.038

✦ Synopsis

It is known that the max-algebraic powers A r of a nonnegative irreducible matrix are ultimately periodic. This leads to the concept of attraction cone Attr(A, t), by which we mean the solution set of a two-sided system λ t (A)A r ⊗ x = A r+t ⊗ x, where r is any integer after the periodicity transient T(A) and λ(A) is the maximum cycle geometric mean of A. A question which this paper answers, is how to describe Attr(A, t) by a concise system of equations without knowing T(A). This study requires knowledge of certain structures and symmetries of periodic max-algebraic powers, which are also described. We also consider extremals of attraction cones in a special case, and address the complexity of computing the coefficients of the system which describes attraction cone.