Inequalities On Generalized Geometric Mean Of Positive Definite Matrices

Matrix theory is an active and important area. It has been fundamental tools in mathematical disciplines.There are extensive applications in the fields of differ-ential equations, probability and statistics, optimization, application in theoretical and applied economics, the engineering disciplines, operations research. Some important conclusions in matrix theory are appeared with forms of matrix in-equations. So matrix inequalities in the research and development of the theory of matrices are significant. matrix inequalities involve matrix norms,singular value, eigenvalues, partial order, numerical range, spectral radius, nonnegative matrices and symbols pattern,etc. The main content of this thesis are as follows:(i) In Chapter 1, we first look back the evolvement of matrix geometric mean. And then introduce the inequalities of generalized geometric mean, generalized harmonic mean, binomial mean on positive definite matrices that we study.(ii) In Chapter 2, we study inclusion intervals of eigenvalues and singular values. In [8], R.A. Horn and C.R. Johnson Introduce some important conclu-sions on inclusion intervals of the matrix eigenvalues and the singular values. We presented modified Brauer-type forms on the eigenvalues and the singular values with partial absolute row sums and column sums.(iii) In Chapter 3, we study some properties of the unitarily invariant norm, and presented a norm inequality for sums and differences of positive matrices.