eigenvalue

Definition

Let $V$ be a vector_space over a field $\mathbb{F}$, and let $T: V \to V$ be a linear operator. A scalar $\lambda \in \mathbb{F}$ is called an eigenvalue of $T$ if there exists a nonzero vector $v \in V$ such that:

$$ T(v) = \lambda v $$

The vector $v$ is called an eigenvector of $T$ corresponding to $\lambda$.

Matrix Form

For a square matrix $A \in \mathbb{F}^{n \times n}$, $\lambda$ is an eigenvalue if and only if:

$$ \det(A - \lambda I) = 0 $$

This equation is called the characteristic equation of $A$.

Eigenspace

The eigenspace of $T$ corresponding to eigenvalue $\lambda$ is the set:

$$ E_\lambda = \{v \in V : T(v) = \lambda v\} = \ker(T - \lambda I) $$

This is a subspace of $V$.

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