Is $\alpha: v \to iv$ normal?

Question

Let n be a positive integer and let $\alpha$ be the endomorphism of $\mathbb{C}^n$ defined by $\alpha: v \to iv$. Is $\alpha$ normal?

I guess, yes. I just want to make sure that I've made a valid argument.

Let $v,\:w\in \mathbb{C}^n$. Hence $\langle \alpha(v),w\rangle=\langle iv,w\rangle =\langle v,-iw\rangle $ which implies that $\alpha^*(v)=-iv$ for all $v\in V.$

Thus, $\alpha^*\alpha(v)=-i(iv)=v=\alpha\alpha^*(v)$ and this means $\alpha$ is normal.

Answer 1

Assuming the inner product is linear in the second variable: $$ \langle \alpha(v),w\rangle= \langle iv,w\rangle= -i\langle v,w\rangle =\langle v,-iw\rangle $$ so indeed $\alpha^*(w)=-iw$. (Similarly if you use the convention that the inner product is linear in the first variable.)

Hence $$ \alpha^*(\alpha(v))=-i^2v=v,\qquad \alpha(\alpha^*(v))=-i^2v=v $$ Good job.