Normal Operator

A more generalized form of Unitary

Definition

• With $V$ as a finite dimension Inner Product Space
• With $T\in \mathcal{L}(V)$
• Then, $T{T}^{\ast }=T^{\ast }T=aI,a\in \mathbb{F}$

Properties

• $||T\alpha ||=||T^{\ast }\alpha ||$
• For scalar $\alpha ,(T-\alpha I)$ is also Normal
• $(T-\alpha I{)}^{\ast }=T^{\ast }-\overline{\alpha }I$
• $T$ has a Eigenvector with Eigenvalue $c$ $⟺$ $T^{\ast }$ has a Eigenvector with Eigenvalue $\overline{c}$

Proofs

• Proving Normal Operator Property 4

Concepts

• Normal Matrix

Theorem

• Operator is Normal if Minimal Polynomial has Factors of Degree 1