Sesquilinear form/Orthogonalizable/Correspondence/Normal/Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, with a fixed inner product ${}\left\langle -,-\right\rangle$. We call a sesquilinear form ${}\Psi$ on ${}V$ orthogonalizable if there exsts an orthonormal basis ${}u_{1},\ldots ,u_{n}$ (with respect to the inner product) of ${}V$ fulfilling

${}\Psi (u_{i},u_{j})=0\,$

for all ${}i\neq j$. Show that via the correspondence

$\operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {Sesq} _{}{\left(V\right)},\varphi \longmapsto \Psi _{\varphi },$

the normal endomorphisms correspond to the orthogonalizable sesquilinear forms.

Create a solution