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Linear Algebra Unit 6: Eigenvalues & eigenvectors

Eigenvectors are directions a matrix only stretches. Eigenvalues are the stretch factors. They unlock diagonalization and long-run behavior.

Covers: Characteristic polynomial, diagonalization, applications (Markov chains, DEs)

Study guide

An eigenvector of a square matrix A is a nonzero vector v with Av = λv: A only stretches it by the factor λ, the eigenvalue. To find eigenvalues, solve the characteristic equation det(A − λI) = 0, a polynomial of degree n. For each eigenvalue, find eigenvectors by solving (A − λI)v = 0; the solutions form the eigenspace. The eigenvalues of a triangular matrix are its diagonal entries, their sum is the trace, and their product is the determinant. A is invertible exactly when 0 isn't an eigenvalue.

A matrix is diagonalizable if A = PDP⁻¹, where D is diagonal with the eigenvalues and P has the matching eigenvectors as columns. This requires n linearly independent eigenvectors. Distinct eigenvalues guarantee it; repeated eigenvalues need eigenspaces of full dimension. Diagonalization makes powers easy: Aᵏ = PDᵏP⁻¹, and Dᵏ just raises each diagonal entry to the kth power. Complex eigenvalues correspond to rotation-like behavior.

Applications: in a Markov chain, a stochastic matrix moves a probability vector from step to step, and the long-run steady state is the eigenvector for eigenvalue 1. In systems of differential equations x′ = Ax, each eigenpair gives a solution e^(λt)v, and the signs of the eigenvalues tell whether solutions grow, decay or oscillate. Dynamical systems xₖ₊₁ = Axₖ behave similarly, with |λ| > 1 meaning growth.

If Av = λv, then the same v is an eigenvector of A² (eigenvalue λ²), Aᵏ (λᵏ), A⁻¹ (1/λ) and A + cI (λ + c). So a matrix with A² = cI can only have eigenvalues ±√c, and the trace tells you how many of each. To compute Aᵏx₀, write x₀ = c₁v₁ + … + cₙvₙ in eigenvectors. Then Aᵏx₀ = c₁λ₁ᵏv₁ + … + cₙλₙᵏvₙ, and terms with |λ| < 1 die out. For a Markov matrix, the limit is the steady-state vector scaled so its entries add up to the same total as x₀'s.

An eigenvalue's algebraic multiplicity is its multiplicity as a root of the characteristic polynomial, and its geometric multiplicity is the dimension of its eigenspace, which is never larger. A matrix whose geometric multiplicity falls short for some eigenvalue is called defective and can't be diagonalized. An example is [2 1; 0 2].

Key ideas

• Av = λv with v ≠ 0.
• Find eigenvalues by solving det(A − λI) = 0.
• Find eigenvectors by solving (A − λI)v = 0.
• A is diagonalizable if it has n independent eigenvectors: A = PDP⁻¹.
• Diagonalization makes powers easy: Aᵏ = PDᵏP⁻¹.
• Trace = sum of eigenvalues; determinant = product of eigenvalues.

Formulas and rules

Common mistakes

• Accepting v = 0 as an eigenvector.
• Assuming every matrix is diagonalizable.

Words to know

Definition of an eigenvector

A nonzero v with Av = λv.

Characteristic equation

det(A − λI) = 0

Diagonalizable matrix

An n×n matrix with n linearly independent eigenvectors.

Powers of a diagonalizable matrix

If A = PDP⁻¹, then Aᵏ = PDᵏP⁻¹.

Trace

Sum of the diagonal entries = sum of the eigenvalues.

Distinct eigenvalues

Guarantee independent eigenvectors, so the matrix is diagonalizable.

Eigenvalue

A scalar λ with Av = λv for some nonzero v.

Eigenspace

All solutions of (A − λI)v = 0.

Eigenvalues of a triangular matrix

Its diagonal entries.

Product of the eigenvalues

det A

Zero eigenvalue

A is not invertible.

Markov chain steady state

The eigenvector for eigenvalue 1 of a stochastic matrix.

Complex eigenvalues

Indicate rotation-like behavior.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.