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
We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and adj(A) det(A) −1 , all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.
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 †
We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a nonsingular matrix A, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue λ, and corresponds to the columns of the eigenmatrix A + λI from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case a certain "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix A ∇ := 1 det(A) adj(A) and the connection of the independence question to generalized eigenvectors.
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