Positive definite matrix problem...

Question

I have studied positive definite matrices . And I came across this exercise.

I can show that A+B is a positive definite matrix by the definition of positive definite matrix

Also in part (2) I can say AB is not possitive definite as it not necessarily symmetric

In part (3) I can conclude that A^2 is positive definite because all its eigenvalues are positive since A is positive definite

But I am not getting idea to proceed in rest of the parts

Please help.

Answer 1

Hint Since $A$ is symmetric, (iv) and (v) coincide, as do (vi) and (vii).

For the former, for any vector $x \in \Bbb R^n$ (where $A, B$ are $n \times n$ matrices) we have $x^{\top} (A^T B A) x = (Ax)^{\top} B (Ax)$.

For the latter, as Paul pointed out in the comments, these quantities both coincide with $A^2$.