Definiteness of square of a positive definite matrix

Question

If $A$ is positive definite ,($\mathbf x^ \mathbf H A \mathbf x> 0$) then can we say that $A^2 $ is also positive definite?

Answer 1

$A$ is positive definite $\iff A$ is the Gram matrix of linearly independent vectors. Suppose this representation is given by $A=V^HV$. Clearly $A^H=A$. $$ A^2=A^HA\implies x^HA^2x=x^HA^HAx=(Ax)^HAx=||Ax||^2\ge0 $$