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

摘要

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A(del), defined as 1/det(A) adj(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(del)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. (C) 2015 Elsevier Inc. All rights reserved.