Unitary vector space/Isometry/Eigenvalues/Fact/Proof
Proof
Let with , that is, is an eigenvector for the eigenvalue . Due to the isometry property, we have
Because of , this implies . In the real case this means .
Let with , that is, is an eigenvector for the eigenvalue . Due to the isometry property, we have
Because of , this implies . In the real case this means .