Alexander E. Guterman

Some general techniques on linear preserver problems

Abstract Several general techniques on linear preserver problems are described. The first one is based on a transfer principle in Model Theoretic Algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0. The second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied. The third one is about solving linear preserver problems on more general (operator) algebras by reducing the problems to idempotent preservers. Numerous examples will be given to demonstrate the proposed techniques.

TROPICAL POLYHEDRA ARE EQUIVALENT TO MEAN PAYOFF GAMES

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.

RANK INEQUALITIES OVER SEMIRINGS

Inequalities on the rank of the sum and the product of two matrices over semirings are surveyed. Preferences are given to the factor rank, row and column ranks, term rank, and zero-term rank of matrices over antinegative semirings. During the past century a lot of literature has been devoted to in- vestigations of semirings. Brie∞y, a semiring is essentially a ring where only the zero element is required to have an additive inverse. Therefore, all rings are also semirings. Moreover, among semirings there are such combinatorially interesting systems as the Boolean algebra of subsets of a flnite set(with addition being union and multiplication being intersec- tion), nonnegative integers and reals(with the usual arithmetic), fuzzy scalars(with fuzzy arithmetic), etc. Matrix theory over semirings is an object of much study in the last decades, see for example (9). In particu- lar, many authors have investigated various rank functions for matrices over semirings and their properties, see (1, 3, 6, 7, 8, 12) and references there in. There are classical inequalities for the rank function ‰ of sums and products of matrices over flelds, see, for example (10, 11): The rank-sum inequalities:

Linear preservers for matrix inequalities and partial orderings

We characterize the linear preservers of minus partial order on matrix algebras. The developed approach allows us to classify linear transformations that live fixed several matrix relations arising as extremal cases in some classical matrix inequalities, including rank-additivity relation and related properties. Applications to the determinant preservers are considered.

LINEAR PRESERVERS FOR DRAZIN STAR PARTIAL ORDER

We characterize the linear preservers of Drazin star partial order for matrix algebras. The linear preservers for left and right star partial orderings are classified also.

Monotone Linear Transformations on Matrices Are Invertible

ABSTRACT We show that monotone linear transformations on matrices with respect to ¯<, , * <, < *, , , -partial orders are invertible and provide a complete characterization of such transformations.

Tropical Cramer Determinants Revisited

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.

Commuting graphs and extremal centralizers

We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra M n ( F ) over an arbitrary field F . It is known that if F is an algebraically closed field and n  ≥ 3 , then the diameter of the commuting graph of M n ( F ) is always equal to four. We construct a concrete example showing that if F is not algebraically closed, then the commuting graph of M n ( F ) can be connected with the diameter at least five.

On the Polya permanent problem over finite fields

Let F be a finite field of characteristic different from 2. We show that no bijective map transforms the permanent into the determinant when the cardinality of F is sufficiently large. We also give an example of a non-bijective map when F is arbitrary and an example of a bijective map when F is infinite which do transform the permanent into the determinant. The technique developed allows us to estimate the probability of the permanent and the determinant of matrices over finite fields having a given value. Our results are also true over finite rings without zero divisors.

Preserving Zeros of a Polynomial

We study nonlinear surjective mappings on ℳ n (𝔽) and its subsets, which preserve the zeros of some fixed polynomials in noncommuting variables.