= Solution
For the common eigenvector $v$, its orthogonal complement $v^\perp$ is invariant under both $\alpha$ and $\alpha^*$. Indeed, if $w\perp v$, then
$$
\langle\alpha w,v\rangle
=\langle w,\alpha^*v\rangle=0,
$$
and the analogous calculation applies to $\alpha^*w$. The restriction of $\alpha$ to $v^\perp$ is therefore again normal. Induction on $\dim V$, beginning with the normalized vector $v/\|v\|$, produces an orthonormal eigenbasis. This proves the finite-dimensional <spectral theorem for normal operators>.
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