Let $V$ be an inner product space and let $\alpha \in End(V )$ be positive definite. Is $α^2$ necessarily positive definite?
I can say that $\alpha^3$ is positive definite, as $\alpha$ is positive definite, but I have no idea how to show $\alpha^2$ is positive definte or if it's not, nothing came to my mind as a counterexample! I appreciate any help.
As, $\alpha$ is positive definite, so it's self-adjoint and for any nonzero vector $v\in V$, we have $\langle \alpha(v),v\rangle>0$.
Since $\alpha^3$ is self-adjoint and for any given $0\neq v\in V$, we have $$\langle\alpha^3(v),v\rangle=\langle\alpha^2(v),\alpha(v)\rangle=\langle\alpha(\alpha(v)),\alpha(v)\rangle>0$$