Groups And Linear Preserving Operators Of Tropical Matrices

The theory of tropical algebra is an important branch of algebra developed over the tropical semiring.It has been widely used in areas such as analysis of discrete event systems,control theory,combinatorial optimisation and scheduling,formal language and automata theory,statistical inference and algebraic geometry,etc.The multiplicative groups of matrices over tropical semiring are the mainly research objects of the tropical algebra theory.In this thesis we introduce the generalized centralizer groups and the centralizer groups of tropicaln× nmatrices,and study the properties of elements of the generalized centralizer groups and of the centralizer groups,and give the structural descriptions of the generalized centralizer groups and the transitive centralizer groups of special matrices,respectively.Moreover,the study of linear preserver problem of tropical matrices is a hot research topic.In this thesis we study the linear operators preserving the weak transitive closures and the strong transitive closures of matrices over tropical semiring or over its subsemiring,respectively.The main contents are as follows:1.We introduce the generalized centralizer groups of tropical n× nmatrices and prove that the generalized centralizer group of a tropical n× nmatrix is a product of its two specific normal subgroups.On this bases,we prove that the generalized centralizer group of a tropical n× nmatrix with connected weighted digraph is a inner product of its two normal subgroups.2.We introduce the centralizer groups of tropical n× nmatrices and prove that every centralizer group of a tropical n×nmatrix is isomorphism to a 2-closed permutation group.Moreover,a finite permutation group is 2-closed if and only if it is isomorphism to the centralizer group of an idempotent normal matrix.Also,the structural descriptions of transitive centralizer groups of idempotent normal matrices which are not strongly regular are given.For a strongly regular idempotent normal matrix with transitive centralizer group,the form of such a matrix is investigated when its centralizer group is equal to the centralizer group of an idempotent normal matrix which is not strongly regular.3.As a continuation of the above,the Frobenius normal form of an idempotent normal matrix is described.And then the transitive centralizer group of a reducible idempotent normal matrix is characterized.For an irreducible idempotent normal matrix with a transitive centralizer group,the form of such a matrix is investigated when its centralizer group is equal to the centralizer group of a reducible idempotent normal matrix.4.We study the linear preserver problem of tropical matrices.Firstly,we characterize the invertible linear operators preserving the weak transitive closures and the strong transitive closures of matrices over tropical semiring or over its subsemirings,respectively.Secondly,we characterize the invertible linear operators strongly preserving the weak transitive closures and the strong transitive closures of matrices,respectively.Finally,we prove that the linear operators strongly preserving the weak transitive closures of n× nmatrices over the binary boolean semiring are invertible when n2,and that the linear operators strongly preserving the strong transitive closures of n× nmatrices over the binary boolean semiring are invertible when n= 2.We also illustrate that there exist non-invertible linear operators that preserve the strong transitive closures when n= 2.