Reducible Spectral Theory with Applications to the Robustness of Matrices in Max-Algebra

Let a b = max(a; b) and a b = a + b for a; b 2 R := R [ f 1g. By max-algebra we understand the analogue of linear algebra developed for the pair of operations ( ; ), extended to matrices and vectors. The symbol A stands for the k max-algebraic power of a square matrix A. Let us denote by " the max-algebraic "zero" vector, all the components of which are 1: The max-algebraic eigenvalue-eigenvector problem is the following: Given A 2 R , …nd all 2 R and x 2 R; x 6= " such that A x = x: Certain problems of scheduling lead to the following question: Given A 2 R , is there a k such that A x is a maxalgebraic eigenvector of A? If the answer is a¢ rmative for every x 6= " then A is called robust. First, we give a complete account of the reducible max-algebraic spectral theory and then we apply it to characterize robust matrices. AMS classi…cation: 15A18