Definition:Diagonalizable Operator

Definition
Let $H$ be a Hilbert space over $\Bbb F \in \left\{{\R, \C}\right\}$.

Let $A:H \to H$ be a linear operator on $H$.

Then $A$ is said to be diagonalizable iff there exist:


 * a basis $E$ for $H$
 * a collection $\left({\alpha_e}\right)_{e \in E} \subseteq \Bbb F$ of scalars (with $E$ as indexing set)

such that:


 * $\forall e \in E: Ae = \alpha_e e$

Value Set
The collection $\left({\alpha_e}\right)_{e \in E}$ may be called the value set of $A$ (with respect to the basis $E$).

Examples

 * Orthogonal Projection