Let $V$ be an inner product space generated over $\mathbb{C}$ and $B$ is a $n\times n$ normal complex matrix.
(1)I need to show that there exists a matrix $C$ such that $C^{2}=B$. I know that B is orthogonally diagonalizable by a theorem which was proved in class. Could I should that $C=BC^{-1}$?
(2) If the eigenvalues of $B$ are real, then $B$ is self-adjoint. Not sure where to start on this one.