Eigenvectors

A precise guide to eigenspaces, nonzero-vector conditions, geometric multiplicity, and invariant directions.

A linear transformation moves an entire grid while preserving lines and the origin. A square source grid maps to a skewed image grid. Basis vectors and one eigen-direction are highlighted, while a signed area tile represents the determinant’s scale factor.

Precise definition

For an eigenvalue $\lambda$ of $A$, an eigenvector is a nonzero vector $v$ satisfying $Av=\lambda v$. The eigenspace $E_\lambda=\ker(A-\lambda I)$ includes zero but its nonzero members are the eigenvectors associated with $\lambda$.

Notation and mathematical language

Solve the homogeneous system $(A-\lambda I)v=0$ to find a basis of $E_\lambda$. Its dimension is geometric multiplicity. Scaling an eigenvector by any nonzero scalar gives another eigenvector for the same eigenvalue.

Conceptual picture

An eigenvector's line is invariant: applying $A$ keeps the result on that line. A basis of eigenvectors decouples a transformation into independent scaling modes. For symmetric real matrices, eigenspaces of distinct eigenvalues are orthogonal.

Fully worked example

Interpretation and application

Eigenvectors identify steady patterns, principal axes, modes, and ranking directions. Their coordinates depend on basis and scale; normalization is a convention unless length has a specific meaning.