Hermetian

Let be a Hilbert Space and be a Linear Operator.
Then is Hermetian or self-Adjoint if
i.e. if

Proposition

For a complex Hilbert Space and a Linear Operator ,
there exist unique Hermetian with .

Proof

Properties

• Eigenvalues of a Hermetian operator are real.
• Two eigenvectors with different eigenvalues of a Hermetian operator are orthogonal.
• The set of all eigenvectors of a Hermetian operator forms a complete orthogonal set.