Supertropical matrix algebra

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11] . Our main results are as follows: The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. There exists an adjoint matrix adj( A ) such that the matrix A adj( A ) behaves much like the identity matrix (times | A |). Every matrix A is a supertropical root of its Hamilton-Cayley polynomial f A . If these roots are distinct, then A is conjugate (in a certain supertropical sense) to a diagonal matrix. The tropical determinant of a matrix A is a ghost iff the rows of A are tropically dependent, iff the columns of A are tropically dependent. Every root of f A is a “supertropical” eigenvalue of A (appropriately defined), and has a tangible supertropical eigenvector.