Research On Numerical Characteristic Inequalities For Sector Matrices

Matrix analysis is an important research area of numerical algebra and its applications.Matrix inequality is an interesting research direction in matrix theory.It focuses on the relations between the sizes of matrices.The relations are concerned with numerical characteristic inequalities mainly including eigenvalues,singular values,norms and so on.Thus,various forms of numerical characteristic inequalities come into being.Basing on the previous numerical characteristic inequalities,the thesis will present some refined results by alternative proof approach and some neglected conclusions.This paper mainly consider unitarily invariant norms inequalities for sector matrices,generalization of mixed arithmetic-harmonic-geometric mean inequalities,determinantal inequalities for a special class of sector matrices-accretive-dissipative matrices and new proof methods and so on.The structure of this thesis is as follows:Some basic definitions of matrix theory and theorem related to the thesis,the background of matrix inequalities and the overview of our results are introduced at first.Secondly,unitarily invariant norms inequalities for sector matrices are studied and the Schatten p norm inequalities are extended to the sums of n(n≥ 2)sector matrices by Cartesian(Hermitian)decomposition and arithmetic-geometric mean inequality.The results are the generalizations of the unitarily invariant norms inequalities for positive semidefinite matrices.In addition,a singular value inequality for sector matrices which refines a known result will be given by the famous Fan-Hoffman inequality.Then the new proofs of two determinant inequalities for Accretive-dissipative matrices by Minkowski inequality and the properties of concave functions are given in chapter three.At the end,the inequality relations between the sector matrix and its real part in Cartesian decom-position by using the properties of convex function and definition and properties of arithmetic-geometric-harmonic-mean are presented firstly.Next,the mixed arithmetic-geometric-harmonic-mean inequalities for sector matrices are proved which are the generalizations of the known results.Finally,the results are extended to sums of n(n≥ 2)sector matrices.