We naturally define the arithmetic mean of two positive definite matrices $A$ and $B$ as $M_A = \frac{1}{2}(A + B)$ and their geometric mean as $M_G = A^{\frac{1}{2}} \left(A^{-\frac{1}{2}} B A^{-\frac{1}{2}} \right)^{\frac{1}{2}} A^{\frac{1}{2}}.$ In the thesis, we will prove essential properties of the geometric mean and we will show that $M_G \leq M_A,$ which means that the matrix $M_A-M_G$ is positive semidefinite. For complex matrices, we will explore and prove two versions of the inequality. The first holds for $A, B \in \mathbb{C}^{n \times n}$ and states
$$s_j(A^* B) \le \frac{1}{2} s_j(AA^* + BB^*),$$
where $s_j$ denotes $j$-th largest singular value of the matrix, $j = 1, \dots, n.$ The second version holds for arbitrary matrices $ A, B, X$ and states $$ s_1(A^*XB) \leq \tfrac{1}{2} s_1(AA^*X + XBB^*). $$
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