Eigenvalues and Eigenvectors

Summary

An eigenvector of a square matrix is a nonzero vector mapped to a scalar multiple of itself. Eigenvalues are those scalars. The spectral picture diagonalizes (or Jordan-reduces) linear maps and underpins differential equations, PCA, and stability analysis.

Prerequisites

Determinants, Vector Spaces and Bases, Matrices and Row Reduction. Hub: Linear Algebra.

Definition / Statement

Let $A \in R^{n \times n}$ (or $C^{n \times n}$ ). A scalar $λ$ is an eigenvalue of $A$ if there exists a nonzero vector $v$ (an eigenvector) such that

A v = λ v .

Equivalently,

(A - λ I) v = 0, v \neq 0,

so $A - λ I$ is singular:

det (A - λ I) = 0.

The polynomial $p_{A} (λ) = det (A - λ I)$ is the characteristic polynomial (degree $n$ ).

Objects and Dimensions

Object	Meaning	Dimensions
$A$	square matrix	$n \times n$
$λ$	eigenvalue	scalar
$v$	eigenvector	$n \times 1$ , nonzero
$E_{λ} = Nul (A - λ I)$	eigenspace	subspace of $R^{n}$ or $C^{n}$

Notation

Symbol	Meaning
$A v = λ v$	eigen-equation
$p_{A} (λ) = det (A - λ I)$	characteristic polynomial
algebraic multiplicity	multiplicity of $λ$ as a root of $p_{A}$
geometric multiplicity	$\dim E_{λ}$

Conditions / Assumptions

Eigenvectors are nonzero by definition; $0$ is never called an eigenvector.
Over $R$ , eigenvalues may be complex (come in conjugate pairs for real $A$ ).
$A$ is diagonalizable if and only if there is a basis of $R^{n}$ (or $C^{n}$ ) consisting of eigenvectors, equivalently the sum of geometric multiplicities equals $n$ (over an algebraically closed field, algebraic and geometric multiplicities must match for each eigenvalue).
Geometric multiplicity is at most algebraic multiplicity and at least $1$ for each eigenvalue.

Matrix / Vector Form

Characteristic equation

det (A - λ I) = 0.

Diagonalization: if $A = P D P^{- 1}$ with $D = diag (λ_{1}, \dots, λ_{n})$ and columns of $P$ eigenvectors, then

A^{k} = P D^{k} P^{- 1}

for integers $k \geq 0$ (and for negative $k$ if $A$ is invertible, i.e. no zero eigenvalue).

Trace and determinant (counting algebraic multiplicity):

tr (A) = \sum_{i} λ_{i}, det (A) = \prod_{i} λ_{i} .

Geometric Interpretation

On the line spanned by $v$ , $A$ acts as scaling by $λ$ .
$λ = 1$ : fixed directions; $λ = - 1$ : flips; $| λ | < 1$ : contraction along that axis (in iterative maps).

Worked Example

A = (\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}) .

A - λ I = (\begin{matrix} 2 - λ & 1 \\ 1 & 2 - λ \end{matrix}), det (A - λ I) = (2 - λ)^{2} - 1 = (λ - 1) (λ - 3) .

Eigenvalues $λ = 1$ and $λ = 3$ .

For $λ = 1$ : $(A - I) v = 0$ gives $v_{1} + v_{2} = 0$ , so $v = ⟨ 1, - 1 ⟩$ .

For $λ = 3$ : $(A - 3 I) v = 0$ gives $- v_{1} + v_{2} = 0$ , so $v = ⟨ 1, 1 ⟩$ .

Common Mistakes

Writing $det (A - λ I) = 0$ as $det (λ I - A) = 0$ without tracking sign: both are fine if used consistently ( $p (λ) = det (λ I - A)$ is monic).
Allowing $v = 0$ as an eigenvector.
Assuming every real matrix has a full set of real eigenvectors.
Confusing algebraic and geometric multiplicity.

Connections

Related: Determinants, Vector Spaces and Bases, Orthogonality and Projections (symmetric matrices: real orthonormal eigenbases)
Next: Orthogonality and Projections, Least Squares and QR
Applications: spectral methods, PCA, differential systems

References

Eigenvalue theory as in MIT 18.06.^[1]

MIT OpenCourseWare, 18.06 Linear Algebra, https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ ↩︎