On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A∇A∇, defined as 1det⁡(A)adj(A), with det⁡(A)det⁡(A) being the tropical permanent (also called the tropical determinant) of a matrix A, inherits some classical algebraic properties and has some surprising new ones. Defining B   and B′B′ to be tropically similar if B′=A∇BAB′=A∇BA, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof of the identity for det⁡(AB)det⁡(AB) and a connection to stabilization of the powers of definite matrices.