Eigenvalues and eigenvectors

An eigenvector is a nonzero vector whose direction is preserved by a square linear map. Its eigenvalue is the corresponding scaling factor.

Ask the Linear algebra assistant1 min read · Updated September 9, 2026

Definition

An eigenvector is a nonzero vector whose direction is preserved by a square linear map. Its eigenvalue is the corresponding scaling factor.

Most vectors rotate or mix under a map; eigenvectors lie along special invariant directions.

For diag(2, 3), (1, 0) has eigenvalue 2 and (0, 1) has eigenvalue 3.

$Av=\lambda v$ with $v\neq0$ . Not every real matrix has a real eigenbasis.

Eigenvalues describe dynamical stability, principal directions of symmetric matrices, and graph properties.