Eigenvalues & Eigenvectors

Eigenvectors are directions a matrix only stretches, and eigenvalues are the stretch factors — the key to understanding repeated application of a linear map.

1. The idea

A nonzero vector $v$ is an eigenvector of $A$ with eigenvalue $λ$ if $A v = λ v$ : applying $A$ to $v$ does not rotate it, only scales it. Eigenvalues solve the characteristic equation $det (A - λ I) = 0$ .

For a symmetric matrix (like any covariance matrix), eigenvalues are always real and eigenvectors corresponding to distinct eigenvalues are orthogonal, which lets us diagonalize $A = Q Λ Q^{T}$ with $Q$ orthogonal.

Eigenvalues govern long-run behavior: repeated multiplication $A^{k} v$ is dominated by the eigenvector of largest $∣ λ ∣$ , which is why the largest eigenvalue drives PCA and stability analysis.

2. Why it matters

Risk models decompose variance along eigenvectors of the covariance matrix, and stability of dynamic systems (e.g. autoregressive processes) is determined entirely by eigenvalue magnitudes.

3. Core mathematics

Eigen-equation

A v = λ v, v \neq = 0

Characteristic polynomial

det (A - λ I) = 0

Trace and determinant

tr (A) = i \sum λ_{i}, det (A) = i \prod λ_{i}

Spectral theorem

A = Q Λ Q^{T}

for symmetric $A$ , $Q$ orthogonal

4. How to recognize it

Think of this when…

Questions asking for 'principal directions' or 'dominant mode'
Stability of a recursion or Markov chain (eigenvalues of the transition matrix)
Diagonalizing a covariance matrix or computing $A^{k}$

5. Worked example

Worked example

Find the eigenvalues and eigenvectors of $A = (2112)$ .

1
Set up the characteristic equation.
$det (A - λ I) = (2 - λ)^{2} - 1 = 0$
2
Solve for eigenvalues.
$(2 - λ)^{2} = 1 \Rightarrow λ = 1, 3$
3
Find eigenvectors by solving $(A - λ I) v = 0$ .
$λ = 3 : v = (1, 1); λ = 1 : v = (1, - 1)$
Answer: λ = 3 with eigenvector (1,1); λ = 1 with eigenvector (1,-1)

6. Common mistakes

Forgetting eigenvectors are only defined up to scale — any nonzero multiple works
Assuming all matrices are diagonalizable (defective matrices are not)
Confusing eigenvalues of $A$ with singular values, which apply to non-square matrices too

7. Quick check

For a symmetric matrix, eigenvectors for distinct eigenvalues are:

The sum of the eigenvalues of $A$ equals:

8. Practice

5 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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