Show $\alpha$ is selfadjoint. Question: Let $V$ be an inner product space finitely generated over $\Bbb C$ and let $\alpha$ be an endomorphism of $V$ satisfying $\alpha \alpha^* = \alpha^2$. Show that $\alpha$ is selfadjoint.
**Edited after more thought:
I know that in order to be selfadjoint $\alpha^*=\alpha$. Becuase this inner product space is over $\Bbb C$ does it change the proof? 
Or, can I show selfadjoint for $\Bbb R$ and then state the proposition that for an inner product space over $\Bbb C$ and an End$(V)$, If $\langle{\alpha(v), v}\rangle \in \Bbb R$ for all $v \in V$, then $\alpha$ is selfadjoint.
 A: Let $v,\:w\in V$, and so we have,
$$ \big\langle (\alpha^*)^2(v), w\big\rangle=\big\langle v,\alpha^2(w) \big\rangle=\big\langle v,\alpha\alpha^*(w)\big\rangle,$$
which implies that $\alpha^2=(\alpha^*)^2=\alpha\alpha^*$. Now we claim that $\alpha\alpha^*=\alpha^*\alpha$.
This is because
 $$\big\langle(\alpha^*\alpha-\alpha\alpha^*)(v),(\alpha^*\alpha-\alpha\alpha^*)(v)\big\rangle=$$
$$\langle\alpha^*\alpha(v),\alpha^*\alpha(v)\rangle+\langle\alpha\alpha^*(v),\alpha\alpha^*(v)\rangle-\langle\alpha^*\alpha(v),\alpha\alpha^*(v)\rangle-\langle\alpha\alpha^*(v),\alpha^*\alpha(v)\rangle=0$$
Thus,
 $$\alpha\alpha^*=\alpha^*\alpha.$$
Now, as
$$\big{\langle}(\alpha-\alpha^*)(v),(\alpha-\alpha^*)(v)\big\rangle=$$
$$\langle\alpha(v),\alpha(v)\rangle+\langle\alpha^*(v),\alpha^*(v)\rangle-\langle\alpha^*(v),\alpha(v)\rangle-\langle\alpha(v),\alpha^*(v)\rangle=0,$$ we get, $\alpha=\alpha^*.$
A: Given $\alpha^*\alpha=\alpha^2$
\begin{align}
\langle{\alpha^*\alpha(v),v}\rangle & =\langle{\alpha^2(v),v}\rangle \\
&=\langle{\alpha(v),\alpha(v)}\rangle \\
&=\langle{v,\alpha\alpha^*(v)}\rangle 
\end{align}
Therefore, $\alpha^*\alpha=\alpha^2=\alpha\alpha^*$ and $\alpha=\alpha^*$ over $\Bbb R$.
Now, since V is an inner product space over $\Bbb C$, $\alpha \in End(V)$ has an adjoint, and we hae shown that if $\langle{\alpha(v), v}\rangle \in \Bbb R$
 for all $v \in V$, then we can say that $\alpha$ is selfadjoint. (Taken from Golan, The linear algebra a beginning graduate student ought to know, Proposition 17.2).
Check and constructive criticism greatly appreciated. 
