*Problem:*
Prove that if $A$ and $AB+BA$ are positive definite matrices, then $B$ is positive definite.
I didn't understand some parts of this problem's solution which is given below:
Let $C=AB+BA$. Now, multiply $C$ from right and left by $A^{-\frac{1}{2}}$ to get: 0< A^{-\frac{1}{2}}CA^{-\frac{1}{2}}=A^{\frac{1}{2}}BA^{-\frac{1}{2}}+A^{-\frac{1}{2}}BA^{\frac{1}{2}}=D+D^*
Where $D=A^{\frac{1}{2}}BA^{-\frac{1}{2}}$
Next, the solution says that it is sufficient to show that $D$ is nonsingular.
My first question: Why is 0< A^{-\frac{1}{2}}CA^{-\frac{1}{2}}, i.e positive definite?
My second question: I can't see why is $D$ being non-singular implies that $B$ is positive definite?