Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 8F / a

The adjoint operator ${\alpha }^{\ast }$ is defined bySince $V$ is a nonzero finite-dimensional complex vector space, $\alpha$ has an eigenvector $v$ with eigenvalue $\lambda$. Normality givesfor every $x$, by expanding both sides and using $\alpha {\alpha }^{\ast }={\alpha }^{\ast }\alpha$. Taking $x=v$ yields

For the common eigenvector $v$, its orthogonal complement $v^{\perp }$ is invariant under both $\alpha$ and ${\alpha }^{\ast }$. Indeed, if $w\perp v$, thenand the analogous calculation applies to ${\alpha }^{\ast }w$. The restriction of $\alpha$ to $v^{\perp }$ is therefore again normal. Induction on $\dim V$, beginning with the normalized vector $v/\Vert v\Vert$, produces an orthonormal eigenbasis. This proves the finite-dimensional spectral theorem for normal operators.