Max-algebraic attraction cones of nonnegative irreducible matrices
The max-algebraic cyclicity theorem states that if \(A\in {\mathbb R}^{n\times n}_+\) is irreducible and its maximal cyclic geometric mean \(\lambda(A)\) equals 1 then the sequence of max-algebraic powers \(A^k\) becomes periodic after some finite transient time \(T(A)\) and that the ultimate period of \(A^K\) is equal to the cyclicity of the critical graph. The cyclicity theorem naturally leads to the concept of attraction cone \(\mathrm{Attr}(A,T)\) which is the solution set of a two-sided system \(\lambda^t (A) A^r \otimes x=A^{r+y}\otimes x\) for any \(r\geq T(A)\). The author finds a complete description of \(\mathrm{Attr}(A,t)\) by a concise system of equations without knowing \(T(A)\). In addition the extremals of such attraction cones are investigated and the computational complexity for finding the coefficients of the system describing the attraction cones is examined.
scientific article; zbMATH DE number 5919622
scientific article