On Anderson-Taylor Type Matrix Inequalities

Matrix theory is a very broad research field in both basic mathematics and applied mathematics.Matrix inequality plays an important role in the study of matrix theory.With the great progress of science and technology,matrix inequality is playing a more and more important role in the fields of probability theory,mathematical statistics,control and computer image processing.In fact,the matrix inequality itself has many problems that deserve further study.This paper studies Anderson-Taylor matrix inequalities on the basis of the existing results.The main work of this paper is as follows:1.An inequality about the sum of positive definite matrices was obtained by using Schur complement,which extended Haynsworth's inequality about positive definite matrices.2.The results of Bergstrom inequality by Fiedler and Markham are studied,the corresponding results are generalized,and the sufficient and necessary conditions for the establishment of the equality are given.3.Combined with the properties of Schur complement of block positive definite matrix,the research results of Lin about Anderson-Taylor matrix inequality are generalized,which can be regarded as a new proof of Lin's results in special cases.The complementarity of Lin's results is discussed by using Schur complement and Arithmetic-Harmonic mean inequality,and the relevant conclusions also provide a lower bound for Olkin's results.4.This paper discusses the result of Olkin's Anderson-Taylor matrix inequality in the sense of exchange matrix,and transforms the problem of positive definite matrix inequality into diagonal matrix by using the relation between diagonal matrix and exchange matrix,and gives the corresponding generalization.5.This paper generalizes the results of Lin's Anderson-Taylor matrix inequality on M-matrices,discusses more general conclusions and puts forward corresponding conjecture.