I am stuck on the following two questions:
My first question:
If $D$ is a non-singular matrix, such that $(D+D^*)$ is positive definite. From my understanding of the problem's solution, it says that $D$ has to be positive definite. Why is this true?
My second question:
If $A$ and $B$ are positive definite matrices, why is : $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$ positive definite? Now the other way: if $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$ positive definite and $A$ is positive definite, does this imply that $B$ is positive definite?