Singular Values ​​and Unitarily Invariant Norm Of The Matrix Inequality

As a basic mathematical tool. Matrix has a wide range of applications in the optimizing field. probability and statistics, numerical analysis and other disciplines. In the theoretical study of the matrix, the singular value inequalities and norm inequalities are very important.By learning matrix and the related courses, and absorbing the basis of previous studies, we mainly study the nature of the matrix singular values and unitarily invariant norms, and then discuss matrix inequalities of the singular values and inequalities of unitarily invariant norms. This article is divided into three chapters, the main contents of each chapter are as follows:In the first chapter, we introduce symbols, definitions, and some of the theorems used in this article. In the first section, we list the singular value matrix, unitarily invariant norm, semi-definite matrix, spectral radius, self-adjoint norm, product of the Hadamard matrix, doubly stochastic matrix and so on; in the second section, we present a number of well-known theorems.In the second chapter, on the basis of the reference, we mainly study the relationship of matrix singular value and eigenvalue, and obtain the correlation inequalities of singular value and eigenvalue, then prove that the conclusions in the literature [3] by using two different methods, list the necessary and sufficient conditions to the establishment of the equal sign.In the third chapter, firstly, we study the nature of unitarily invariant norm. Secondly, by using the nature of unitarily invariant norm and singular value decomposition theorem, we extend some common inequalities of the complex field to the square, and obtain the unitarily invariant norm inequalities of general matrix. Lastly, by constructing semi-definite matrix, we figure out the product of matrices and the unitarily invariant norm inequalities of Hadamard matrix.