Research On Matrix Singular Value And Unitary Invariant Norm

Matrix theory is not only of abundant research content, but also a mathematical tool of practical value. At the study of the matrix theory, singular value and singular value decomposition play an important role on high-quality statistical computations and the theory of unitariy invariant norms. Over the years, many experts and scholars did a lot of research on the problem of singular values by using different methods from different angles, and expanded the application range of singular values and theoretical system. Some new proofs of singular value decomposition and the Weyl’s theorem are obtained in this paper, also, some simple corollaries are generated. At the same time, unitarily invariant norm, of numerical characteristics in matrix theory, has an important role on the matrix calculation, optimal approximation and disturbance theory. In addition, as a special matrix, block matrix is an important tool to discuss the intrinsic properties of the matrix. This paper, combining positive semi-definite block matrix and the unitary invariant norm, derives a series of matrix inequality for unitarily invariant norm.This article is divided into three chapters, specific content as follows:The first chapter introduce some symbol and concepts referred in this article, for example normal matrix, unitary matrix, Hadamard product, weak majorizatio, log-weak majorization, contraction, generalized inverse. At the same time, some known conclusions and theorems are also presented.The second chapter firstly provide the theorem of singular value decomposition, then we present some new proofs of singular value decomposition via spectral decomposition and polar decomposition and weak majorization relationship between the main diagonal entries and singular values of a matrix. Several new proofs of Weyl’s theorem and corollary are obtained in the third quarter.The third chapter firstly introduce some characteristics of unitarily invariant norm and block matrix, then obtain a series of matrix inequality for unitarily invariant norm by combining unitarily invariant norm and positive semidefinite 2-by-2 block matrices in the third quarter.