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

Adi Niv (1)

1 MAXPLUS - Max-plus algebras and mathematics of decision

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A ∇ , 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 ∇ 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.

Mots clés: Tropical and supertropical linear algebra; characteristic polynomial; eigen-values; Kleene star; permanent; definite matrices; pseudo-inverse AMSC: 15A09 (Primary); 15A15; 15A18; 15A80; 15B33

Domaines: Mathématiques [math]

Adi Niv. On pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra. Linear Algebra and its Applications, 2015, 471, pp.264-290. ⟨10.1016/j.laa.2014.12.038⟩. ⟨hal-01253421⟩