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

Abstract

International audienceThe 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

Dependence of supertropical eigenspaces

“…The resolution by means of the quasi-inverse. In view of the results in [6], [28] and [29], one can conclude that quasi-inverse matrices play an important role in formulating properties of matrices. These studies lead us to the following two conjectures, based on a further examination of Example 3.3. increases the coefficient of x, causing 47x 2 to be inessential.…”

“…The following lemma has been proved in [28,Lemma 3.2], and states the connection between multiplicity of the determinant and the quasi-inverse matrix: Lemma 2.17. Let P be an invertible matrix and A be nonsingular.…”

Tropical compound matrix identities

“…In contrast with the situation over rings, the invertibility of the determinant does not imply that a matrix is invertible. Nevertheless, as shown in several works, especially [Plu90], [Niv15], the familiar expression…”

On the characteristic polynomial of a supertropical adjoint matrix

“…Let (G, * , 0, ≤) be an ordered Abelian group, and G (0) and G (1) be two copies of G. We consider the semiring S = G (0) ∪ G (1) ∪ {ε} with two commutative operations, denoted by ⊕ and ⊙. Assume i, j ∈ {0, 1}, s ∈ S, a, b ∈ G, and let a < b; the operations are defined by ε (ij) .…”

“…Assume i, j ∈ {0, 1}, s ∈ S, a, b ∈ G, and let a < b; the operations are defined by ε (ij) . One can note that S is isomorphic to supertropical semifield, and the elements from G (0) and G (1) correspond to ghost and tangible elements, respectively. We define the mapping ν sending a (i) to a ∈ G and ε to ε; we say that c, d ∈ S are ν-equivalent whenever ν(c) = ν(d), and we write c ≈ ν d in this case.…”