On matrix powers in max-algebra
The max-algebra, an analogue of classical linear algebra, is developed over the semiring \(\bar{\mathbb R}:= \mathbb R \cup \{- \infty\}\) (alias \(\mathbb R _{\max}\)) with respect to the max-plus operators \(a \oplus b= \max(a, b)\) and \(a\otimes b= a + b\) for all \(a, b \in \bar{\mathbb R}\). A square matrix \(A= (a_{ij}) \in \bar{\mathbb R}^{n \times n}\) is called robust if for every \(x \in \bar{\mathbb R}^n\) with at least one finite component, \(A^{(k)} \otimes x\) is an eigenvector of \(A\) for some natural number \(k\), \(A^{(k)}\) being the max-algebraic power of \(A\). By studying the eigenvalue-eigenvector problem for powers of irreducible matrices, the authors characterise robust irreducible matrices. In particular, a strongly irreducible matrix \(A\) is robust if and only if the eigenspaces of all powers of \(A\) coincide.
scientific article; zbMATH DE number 5132604