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

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

Solved by gpt-5.6-sol high.