Some Inequalities On Sector Matrices

Matrix inequality is an important aspect of the matrix theory,which widely occurs in many aspects of pure mathematics,applied mathematics and computational science.For example,operator inequality in functional analysis,spectral theory in graph,pertur-bation analysis in numerical algebra,stability analysis in control theory and so on.These applications in turn also stimulate and promote the further studies of matrix inequalities.In this essay,we investigate a class of special matrices whose numerical range is contained in a sector,commonly referred to as sector matrices.The paper is organized as follows.In chapter 1,we introduce the research background,the progress and the application of the matrix inequalities arising in some specific mathematical problems.In chapter 2,we introduce the symbols used in this article and give some preliminaries which will be used in this paper.In chapter 3,we first derive some determinantal inequalities for the sector matrix.Then we prove an important property of sector matrices on the geometric mean,that is R(A#B)?(RA)#(RB).Finally,we summarize some results on the unitary invariant norm of sector matrices.