Properties Of The Perron Complement For Several Special Matrices

It is well known that Perron complement has many good and special properties and thus can be applied in statistics, computational mathematics, matrix theory and so on. The concept of Perron complement of nonnegative and irreducible ma-trix was introduced by Meyer and it was used by him to construct an algonorithm for computing the stationary distribution vector for Markov chain. As the Perron complement of a nonnegative irreducible matrix is nonnegative irreducible, then, lots of work had done on several special matrices within a nonnegative irreducible matrix. In this paper, we provide several closure properties for Perron comple-ments of diagonally dominant matrices and nekrasov matrices.In chapter one, we first introduce background knowledge and recent research for results the Perron and Schur complement of three known subclasses of H-matrices, and introduce our main works, some basic symbols and definitions.In chapter two, we analyze diagonally dominant degree for the Perron com-plement upon several diagonally dominant cases by using the entries and spectral radius of the original matrix. At the same time, according to some techniques of inequalities, we obtain closure properties for the Perron complement of diagonally dominant matrices, strictly ?-diagonally dominant matrices and strictly product ?-diagonally dominant matrices, which improve and generalize the related results.In chapter three, we get several closure properties for Perron complements of Nekrasov matrices by constructing a new matrix through partitioning the given matrix and applying the relationship between Schur complement of the new con-structed matrix and Perron complement of the original matrix, which improve and generalize the related results.