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

Abstract

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse $A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ 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 $B'$ to be tropically similar if $B' =A^\nabla BA$, 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 $det(AB)$ and a connection to stabilization of the powers of definite matrices.