Core of a matrix in max algebra and in nonnegative algebra: a survey

This paper presents a light introduction to Perron–Frobenius theory in max algebra and in nonnegative linear algebra, and a survey of results on two cores of a nonnegative matrix. The (usual) core of a nonnegative matrix is defined as $\cap_{k\geqslant 1} {\rm span}_+ (A^k)$, that is, intersection of the nonnegative column spans of matrix powers. This object is of importance in the (usual) Perron-Frobenius theory, and it has some applications in ergodic theory. We develop the direct max-algebraic analogue and follow the similarities and differences of both theories.