Factorization of Tropical Matrices

Abstract

In contrast to the situation in classical linear algebra, not every tropically non-singular matrix can be factored into a product of tropical elementary matrices. We do prove the factorizability of any tropically non-singular 2×2 matrix and, relating to the existing Bruhat decomposition, determine which 3 × 3 matrices are factorizable. Nevertheless, there is a closure operation, obtained by means of the tropical adjoint, which is always factorizable, generalizing the decomposition of the closure operation * of a matrix. IntroductionThe tropical semifield is an ordered group G (usually the set of real numbers R or the set of rational numbers Q), together with −∞, denoted as T = G {−∞}, and equipped with the operations a b = max{a, b} and a b = a + b, denoted as a + b and ab respectively (see [1], [9] and [18]). This arithmetic enables one to simplify non-linear questions by answering them in a linear setting (see [7]), which applied in discrete mathematics, optimization, algebraic geometry and more, as has been well reviewed in [3], [4], [5], [6], [8], [16] and [19].This structure can also be studied via the valuation over the field K = C{{t}} of Puiseux series to the ordered group (Q, +, ≥), as has been done in [2], by looking at the dual structure trop(a) = −val(a) denoted as the tropicalization of a ∈ K. In order to make the connection between the results in the work of Buchholz in [2] and the results in this paper we say trop(a + b) = max{trop(a), trop(b)}. Then it is obvious that the tropical structure deals with the uncertainty of a = b in the valuation, in the form of trop(a + a) = trop(a).In this paper we aspire to solve the tropical factorization problem raised in [2]and [20], by passing to a wider structure called the supertropical semiring (see [10] and [11]), equipped with the ghost ideal G = G ν . We denote as R = T G {−∞} the supertropical semiring, where T = G, which contains the so called tangible elements of the structure and ∀a ∈ T, a ν ∈ G are the ghost elements of the structure, as defined in [10]. So G inherits the order of G. We distinguish between a maximal element a that is being attained once, i.e. a ∈ T which is invertible, and a maximum that is being attained at least twice, i.e. a + a = a ν ∈ G, which is not invertible.The work in [12], [13] and [14] shows that even though the semiring of matrices over the supertropical semiring lacks negation, it satisfies many of the classical matrix †

Tropical totally positive matrices

“…Tropical Jacobi elementary matrices are defined in a way analogous to the classical situation, see Section 5. The semigroup they generate was studied in [Niv14]: whereas classically, every nonsingular matrix can be factored in terms of elementary matrices, the same is not true in the tropical setting. Nevertheless, it was shown there that the set of 3×3 tropical matrices which admits such a factorization admits a combinatorial characterization.…”

Algebras with a negation map

“…It contains all the elementary matrices, but is not generated by them multiplicatively, cf. [60]. Just as SL n (A) is a classical algebraic group, with its symmetrized version given in Definition 6.26, we can define PSLM n (A) by taking SLM n (A) modulo the congruence {(A, αA) : α ∈ A}.…”

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

“…Definition 2.9. Following the notation in [26], we define three types of tropical elementary matrices, corresponding to the three elementary matrix operations, obtained by applying one such operation to the identity matrix. A transposition matrix is obtained from the identity matrix by switching two rows (resp.…”

“…As shown in [26] and [28], reductions to definite matrices can simplify verifications of some complicated properties of matrices. On top of that, matrices of this form enjoy some interesting properties of their own.…”