n this thesis we persent matrix form of Heinz inequality and its refinements for matrix invariant norms. Also, we study the reverse of matrix form of Heinz inequality with using scalar Young inequality. Let a and b be nonnegative real numbers. The Heinz means are defined as H amp;#??;(a; b) = a amp;#??;b??? amp;#??; amp;#??; b amp;#??;a??? amp;#??; ? ; ? amp;#??; amp;#??; amp;#??; ? : Let Mn be the space of n amp;#?; n complex matrices and jjj amp;#?;jjj stand for any unitarily invariant norm on Mn, i.e., jjjUAV jjj = jjjAjjj for all A ? Mn and for all unitary matrices U; V ? Mn. For A = [aij ] ? Mn, the Hilbert-Schmidt norm of A is defined by jjjAjjj ? = vuut ?n i;j=? jaij j?: It is known that the Hilbert-Schmidt norm is unitarily invariant. In this thesis, we always suppose that AX ? Mn with A;B positive semidefinite. The well known Heinz mean inequality says that for every positive semidefinite matrices A and B and for every unitarily invariant norm, jjjApBq amp;#??; AqApjjj ? Ap amp;#??;q amp;#??; amp;amp;#??;q where p and q are positive real numbers. A related inequality to the Heinz mean inequality is that jjjApBq amp;#??; BpAqjjj ? Ap amp;#??;q amp;#??; amp;amp;#??;q : Replacing A and B by A ? p amp;#??;q ; B ? p amp;#??;q recepectively, the we get the following equivalent inequality AtB???t amp;#??; BtA???t ? jjjA amp;#??; Bjjj where t ? [?; ?]. If X ? Mn is a Hermitian matrix. then the eigenvalue vector of X is denoted by amp;#?;(X) = ( amp;#??;?(X); amp;#?; amp;#?; amp;#?; ; amp;#??;n(X)) with amp;#??;?(X) amp;#??; amp;#??;?(X) amp;#??; amp;#?; amp;#?; amp;#?; amp;#??; amp;#??;n(X). The singular values of X are the eigenvalues of positive semidefinite matrix jXj = (X amp;#?;X) ? ? ., they are numerated as s?(X) amp;#??; s?(X) amp;#??; amp;#?; amp;#?; amp;#?; amp;#??; sn(X). The vector of the singular values is denoted S(X) = (s?(X); amp;#?; amp;#?; amp;#?; ; sn(X)). Let x = (x?; amp;#?; amp;#?; amp;#?; ; xn) and y = (y?; amp;#?; amp;#?; amp;#?; ; yn) be two vectors in Rn. The vector x is said to be weakly majorized by y and denoted by x ?w y, if and only if ?n j=? x # j amp;#??; ?n j=? y # j ; k = ?; ?; amp;#?; amp;#?; amp;#?; ; n ; where x # ?; amp;#?; amp;#?; amp;#?; ; x # n are the component of x rearranged in decreasing order. It is well known that for every X; Y ? Mn, S(XY ) ?w S(X)S(Y ) : Fan Dominance theorem implies that for every unitary invariant norm S(XY ) ?w S(X)S(Y ) if and only ifis jjjAjjj amp;#??; jjjBjjj : The thesis is structured as follows. In Chapter ?, the basis definitions and theorems has been discussed. In Chapter ?, we study the folowing form og Heinz mean inequality for positive definite batrices A and B as follows AtB???t amp;#??; BtA???t ? ? j ? ? ??tjjjjA amp;#??; Bjjj where t ? [?; ?]. In Chapter ?, by using a Refinements of the scalar Young’s inequality, we introduce an improved Young and Heinz inequalities for matrices. Finally in Chapter ?, the reverses of the classical Young and Heinz inequalities for matrices have been studied.