Cramer and Cayley-Hamilton in the max algebra

Abstract Cramer's rule and the Cayley-Hamilton theorem are formulated and provided in the so-called max algebra, which consists of the set of reals provided with two operations: maximization and addition. It is surprising to see that these well-known theorems carry over to the max algebra almost without any changes, provided that the conventional addition and multiplication are replaced by maximization and addition respectively. The definitions of determinant and eigenvalue have to be adjusted to this new formulation. The role of the determinant is taken over by the permanent and a refinement of this latter concept called the dominant.