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 {divides}A{divides}).• 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.•