Eigenvalues

A rigorous guide to eigenvalues, characteristic polynomials, invariants, and their limits.

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

A scalar $\lambda$ is an eigenvalue of a square matrix $A$ if there exists a nonzero vector $v$ with $Av=\lambda v$. Equivalently, $A-\lambda I$ is singular, so eigenvalues are roots of the characteristic equation $\det(A-\lambda I)=0$.

Notation and mathematical language

The characteristic polynomial has degree $n$ for an $n\times n$ matrix over any field. Over an algebraic closure, it has $n$ roots counting algebraic multiplicity. Those roots sum to $\operatorname{tr}A$ and multiply to $\det A$.

Conceptual picture

An eigenvalue is a scaling factor along an invariant direction. It can be negative, reversing direction, or complex, encoding rotation-scaling in a complexified space. Eigenvalues depend on the linear transformation, not the chosen basis: similar matrices share them.

Fully worked example

Interpretation and application

Eigenvalues determine modes in differential equations, long-term matrix powers, stability, and principal directions for symmetric matrices. A large eigenvalue magnitude need not mean a general nonnormal matrix stretches every vector that much in one step.