This thesis introduces function application on hermitian matrices and proves some of its properties. We define the Löwner partial order for positive semidefinite (psd.) matrices and prove its characterisation with the operator norm and spectral radius. We also introduce (and prove) the operator monotonicity in special cases of matrices, in the case of square root function ($x \mapsto \sqrt{x}$), and in the case of the power function ($x \mapsto x^p$) for exponents in the range $[0,1]$ (Löwner-Heinz inequality), by utilising dyadic numbers. After analyzing the behaviour of the Löwner partial order under matrix inversion we extend the Löwner-Heinz inequality exponent interval to $[-1,1]$. Finally, we prove the Furuta inequalities and demonstrate their relation with the Löwner-Heinz inequality.
|
