(i) Let $V$ be an $n$-dimensional inner-product space over $\mathbb{C}$ and let $\alpha: V \rightarrow V$ be a Hermitian linear map. Prove that $V$ has an orthonormal basis consisting of eigenvectors of $\alpha$.

(ii) Let $\beta: V \rightarrow V$ be another Hermitian map. Prove that $\alpha \beta$ is Hermitian if and only if $\alpha \beta=\beta \alpha$.

(iii) A Hermitian map $\alpha$ is positive-definite if $\langle\alpha v, v\rangle>0$ for every non-zero vector $v$. If $\alpha$ is a positive-definite Hermitian map, prove that there is a unique positivedefinite Hermitian map $\beta$ such that $\beta^{2}=\alpha$.