What is meant by a Hermitian matrix? Show that if $A$ is Hermitian then all its eigenvalues are real and that there is an orthonormal basis for $\mathbb{C}^{n}$ consisting of eigenvectors of $A$.

A Hermitian matrix is said to be positive definite if $\langle A x, x\rangle>0$ for all $x \neq 0$. We write $A>0$ in this case. Show that $A$ is positive definite if, and only if, all of its eigenvalues are positive. Show that if $A>0$ then $A$ has a unique positive definite square root $\sqrt{A}$.

Let $A, B$ be two positive definite Hermitian matrices with $A-B>0$. Writing $C=\sqrt{A}$ and $X=\sqrt{A}-\sqrt{B}$, show that $C X+X C>0$. By considering eigenvalues of $X$, or otherwise, show that $X>0$.