The Singular Values Of Matrix And The Properties Of Contraction

The properties of matrix are studied. The singular values of matrix are inves-tigated and the properties of contraction and its unitary dilation are researched. It is divided into 4 chapters. Here lists some main results.In chapter 1, some notations definitions are introduced and some well-known theorems are given. In section one, we introduce the definitions of the singular values of matrix, positive semidefinite matrix, Schatten-p norm; unitary invariant norms, spectrum norm and so on. In section two, we give some well-known theorems.In chapter 2, we investigate some singular value inequalities of matrix and the conditions for these equalities by using block matrix.In chapter 3, firstly, we discuss some singular values inequalities of positive semidefinite matrix. Subsequently, we investigate some singular values inequalities of semidefinite matrices which are communicate, which gives affirmatively answer to the open problems proposed by Xing-zhi Zhan in [18]. Lastly, we show some inequalities about invariant norms of matrix.In chapter 4, firstly, we investigate the properties of contraction by using spec-trum norm, numerical range. Subsequently, we study some unitary dilation of con-traction by using the method of constructing block matrix.