On Singular Values And Norms Inequalities For Matrices

Matrix inequalities play a key role in many areas such as, mathematical physics,statistics, engineering, operations research, pure mathematics, etc. This field is activeand experiencing a tremendous boost with the passage of time in theory as well as inapplications. Many researchers are working to create an awareness of the theory ofmatrix inequalities. From the last few decades, advancement of functional analysis hasbrought depth in matrix inequalities. Thus the theory of matrix inequalities may beregarded as an independent area of mathematics. This PhD thesis is devoted to specialkind of inequalities namely arithmetic-geometric inequality, Heinz inequalities, Heronand Heinz means inequalities, Cauchy-Schwarz inequalities and absolute valueinequalities and some other related inequalities of matrices. This thesis comprises of sixchapters which briefly describes as under:In Chapter1, we introduce some basic notions and significant results from thetheory of matrix inequalities and briefly review some matrix forms of meansinequalities.In Chapter2, we present some singular value inequalities of matrices by means ofblock matrices techniques. Our results generalize some existing inequalities in theliterature.In Chapter3, we establish new refinements of the well known Hermite-Hadamardinequality. Then as applications of our results, we present several refinements of theHeinz inequalities for matrices by utilizing the convexity of the function on thespecified interval.In Chapter4, we give some unitarily invariant norms inequalities involving Heronand Heinz means for matrices by using the convexity of the function on the specifiedinterval. Our results refine some existing Heron and Heinz means inequalities formatrices.In Chapter5, we obtain some refinements of the matrix Cauchy-Schwarzinequality for unitarily invariant norms by means of convex functions on specifiedinterval, additionally numerical example shows the effectiveness of our results.Finally, in Chapter6, we generalize some norms inequalities for sums, differences,and products of absolute value matrices. Our results based on Minkowski type inequalities and generalized forms of the Cauchy-Schwarz inequality. Furthermore, wediscuss some other related inequalities.