# Definition:Diagonal Operator

## Definition

Let $X$ be a vector space.

Let $\dim X$ be the dimension of $X$.

Let $\set {e_i}_{1 \mathop \le i \mathop \le \dim X}$ be the basis of $X$.

Let $\Lambda : X \to X$ be a mapping such that:

$\forall i \in \N_{> 0} : i \le \dim X : \exists \lambda_i \in \C : \map \Lambda {e_i} = \lambda_i e_i$

Then $\Lambda$ is called the diagonal operator.