Study Of Some Problems Of Sign Pattern Matrices

The matrix theory of a sign patterns matrix studies mainly on the combinatorialproperties that base solely on the signs in the matrix in its qualitative class, regardlessof the magnitudes of its nonzero entries. Sign pattern matrices are well-known to arisein economic as mathematical models-linear dynamic system which are proposed to solvethe problems of international economic by economists P. A. Samuelson. Sign patternmatrices have a very wide range of application backgrounds in the field of economics,chemistry, sociology, and theoretical computer science.The research focuses on the sign pattern in the following directions: sign solvable oflinear equations, sign stable, characterizations of the inertia of sign pattern matrices, com-binatorial characterizations of special sign pattern matrices, the complex generalizationsof sign pattern matrices and so on.Spectrally arbitrary ray patterns are considered. The notation of minimally spectrallyarbitrary ray patterns is introduced. Three families of ray pattern matrices of the formsE_n,3（θ）, F_n,m（θ） and S_n（θ） are given. By the Nilpotent-Jacobi method, it is proved thatthere exist infinitely many choices for θ with0≤θ≤2Ï€, so that E_n,3（θ） and S_n（θ）are spectrally arbitrary and all of their superpatterns are spectrally arbitrary, F_n,m（θ） isspectrally arbitrary and all of their superpatterns are spectrally arbitrary in seven cases.It is also proved that if E_n,3（θ）, F_n,m（θ） and S_n（θ） are spectrally arbitrary, then they areminimally spectrally arbitrary.The period and base of a powerful reducible ray pattern matrix are considered. Twosufficient and necessary conditions for a powerful reducible ray pattern matrix to be peri-odic are given, that is, a powerful reducible ray pattern matrix A is periodic if and only ifthe wight of each cycle in A is periodic, if and only if red（A） is periodic. The period of areducible powerful ray pattern matrix is characterized. Some properties of a non-powerfulray pattern matrix are given.Sign pattern matrices which allow P0-matrix are considered, some sign pattern ma-trices allowing P₀-matrix are given.Quasi-primitive zero-symmetric sign pattern matrices with zero trace are considered.Some properties of a quasi-primitive zero-symmetric sign pattern matrix with zero trace are given. For example, let A be an n order quasi-primitive zero-symmetric （generalized）sign pattern matrix with zero trace if all the2-cycles in S（A） are positive, then l（A）≤2n2, if l（A）=2n-1, then there exits an odd cycle C in S（A） and C is the only cycle inS（A）, all2-cycles in S（A） are negative. The quasi-primitive zero-symmetric sign patternmatrices with zero trace attaining the maximum base2n-1are characterized.Combinatorial properties of a non-negative matrix bases solely on the positions ofentries in the matrix, regardless of the magnitudes of its nonzero entries. Therefore, in thestudy of combinatorial properties of non-negative matrices, a non-negative matrix can betransformed into a （0,1）-matrix. From the viewpoint of sign pattern matrices,（0,1）-matrixcan be regarded as a zero-nonzero pattern matrix. Since an n×n zero-nonzero patternmatrix corresponds to an n order digraph, the study of a zero-nonzero pattern matrix isequivalent to the study of a digraph.Generalized μ-scrambling indices of primitive digraphs are considered. Some exactlower and upper bounds for the generalized μ-scrambling index of various classes ofprimitive digraph are given.