Eigenvector

An eigenvector of a linear transformation is a nonzero vector that changes by only a scalar factor when that transformation is applied.

Definition

Let $T: V \to V$ be a linear operator on a vector_space $V$. A nonzero vector $v \in V$ is an eigenvector of $T$ if there exists a scalar $\lambda$ such that:

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

The scalar $\lambda$ is called the eigenvalue corresponding to $v$.

Matrix Form

For a square matrix $A$, a nonzero vector $v$ is an eigenvector if:

$$ Av = \lambda v $$

Equivalently:

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

where $I$ is the identity matrix.

Properties

• Eigenvectors are only defined up to scalar multiples: if $v$ is an eigenvector, so is $cv$ for any nonzero scalar $c$
• Eigenvectors corresponding to distinct eigenvalues are linearly_independent
• The set of all eigenvectors for a given eigenvalue, together with the zero vector, forms a subspace called the eigenspace

Related

• eigenvalue
• eigenfunction
• linearly_independent
• vector_space