Module 6: Eigenvalues and Eigenvectors

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Learning outcomes

After this module, you should be able to:

• define eigenvalues and eigenvectors and compute them;
• write and solve the characteristic equation;
• interpret eigenvalues/eigenvectors geometrically;
• explain the significance of eigenvalues in applications;
• relate similarity transformations to eigenvalues.

Prerequisites

Complete Module 5 first. Review its quick-revision section if any term below feels unfamiliar.

Short study blocks

Study one row at a time. Do not start diagonalisation until eigenvalues and eigenvectors can be calculated confidently.

Block	Topic	Suggested time
1	Eigenvalue and eigenvector intuition	10-15 minutes
2	Characteristic equation and calculation	10-15 minutes
3	Diagonalisation and geometric interpretation	10-15 minutes
4	Stability, PCA, PageRank, and similarity	10-15 minutes

Start here: the simple idea

An eigenvector of a matrix A is a direction that A only stretches (it is not rotated). The amount of stretch is the eigenvalue. Eigenvalues tell you how a system behaves over time and whether it is stable.

Everyday analogy

Stretch a rubber sheet with a grid drawn on it. Most grid lines bend and rotate, but two special directions only get lengthened (or shrunk) — those are the eigenvectors, and the stretch factors are the eigenvalues.

Eigenvalues and eigenvectors

For an n×n matrix A, a non-zero vector v is an eigenvector of A with eigenvalue λ if

A v = λ v

• v must be non-zero (by definition; λ = 0 with a zero v is trivial).
• λ can be any scalar (including 0 or negative).

The characteristic equation

Rearrange Av = λv ⇒ (A − λI)v = 0. For a non-trivial solution v ≠ 0, the matrix (A − λI) must be singular (non-invertible), i.e.,

det(A − λI) = 0

This is the characteristic equation (a polynomial in λ of degree n). Its roots are the eigenvalues. For each λ, solve (A − λI)v = 0 for v.

Worked example: M = [[2, 1], [1, 2]]

Find eigenvalues

A − λI = [[2−λ, 1    ],
          [1,     2−λ]]

det(A − λI) = (2−λ)² − 1 = λ² − 4λ + 3 = (λ − 3)(λ − 1)

Eigenvalues: λ₁ = 3, λ₂ = 1.

Find eigenvectors

For λ₁ = 3: solve (A − 3I)v = 0 ⇒ [[-1, 1],[1, -1]] [x,y]ᵀ = 0. −x + y = 0 ⇒ y = x. Eigenvector v₁ = (1, 1) (any non-zero multiple works).

For λ₂ = 1: solve (A − I)v = 0 ⇒ [[1, 1],[1, 1]] [x,y]ᵀ = 0. x + y = 0 ⇒ y = −x. Eigenvector v₂ = (1, −1).

Check: A·(1,1) = (3,3) = 3·(1,1) ✔ ; A·(1,−1) = (1,−1) = 1·(1,−1) ✔.

Diagonalisation (significance of eigenvalues)

If A has n linearly independent eigenvectors, form P = [v₁ v₂ … vₙ] and D = diag(λ₁,…,λₙ). Then:

A = P D P⁻¹         (diagonalisation)

Worked example (continued):

P = [[1, 1 ],    D = [[3,0],[0,1]],   P⁻¹ = (1/2)·[[1,1],[1,-1]]
     [1, -1]]

Check: P D P⁻¹ = [[1,1],[1,-1]]·[[3,0],[0,1]]·(1/2)[[1,1],[1,-1]] = [[2,1],[1,2]] = A ✔

When A is diagonalisable, powers are easy: Aᵏ = P Dᵏ P⁻¹.

Geometric interpretation

• Eigenvectors give the principal directions of the transformation.
• Eigenvalues give the scale factors along those directions.
• det(A) = product of eigenvalues = overall area/volume scale.
• det(A − λI) = 0 means those directions are exactly the ones where A acts like scaling (no rotation).

For M = [[2,1],[1,2]]: the transformation stretches the (1,1) direction by 3× and the (1,−1) direction by 1× (unchanged). The "skew square" becomes a rectangle stretched 3× along the main diagonal.

Significance of eigenvalues and eigenvectors

• Discrete-time linear systems: for x_(k+1) = Ax_k, the system decays to zero when every eigenvalue satisfies |λ| < 1; an eigenvalue with |λ| > 1 produces growth in its eigendirection. Continuous-time systems use a different test: the real parts of all eigenvalues must be negative for asymptotic stability.
• PCA (Principal Component Analysis): the axes of greatest data variance are the eigenvectors of the covariance matrix; eigenvalues rank importance.
• Google PageRank: the dominant eigenvector of the link matrix ranks pages.
• Vibration/modes: eigenvectors of a stiffness matrix are the natural vibration modes; eigenvalues are their frequencies².
• PageRank / Markov chains: the steady state is the eigenvector for λ = 1.

Similarity transformation and eigenvalues

Matrices A and B are similar if B = P⁻¹AP for some invertible P. Similar matrices represent the same linear map in different bases.

Key facts:

• Similar matrices have the same eigenvalues (and same det, trace, rank).
• trace(A) = Σ λᵢ, det(A) = Π λᵢ.
• Diagonalisation is exactly the statement that A is similar to a diagonal D.

Common mistakes

• Forgetting that eigenvectors must be non-zero.
• Thinking eigenvalues must be positive (negative and zero are allowed).
• Computing P D P⁻¹ with the wrong order (it's P D P⁻¹, not P⁻¹ D P for the A=PDP⁻¹ form).
• Assuming every matrix is diagonalisable (only if it has n independent eigenvectors).

Memory rules

• Av = λv; solve det(A−λI) = 0 for λ, then (A−λI)v = 0 for v.
• det(A) = product of eigenvalues; trace(A) = sum of eigenvalues.
• Diagonalisation: A = P D P⁻¹; Aᵏ = P Dᵏ P⁻¹.
• Eigenvectors = directions stretched; eigenvalues = stretch factors.
• Similar matrices share eigenvalues and trace/det.

Mini Quiz

Attempt all questions before revealing the answers.

1. How many complex eigenvalues does a 3×3 real matrix have when algebraic multiplicity is counted?
2. If λ is an eigenvalue of A, is λ also an eigenvalue of Aᵀ?
3. True or false: every matrix is diagonalisable.
4. If det(A) = 0, what eigenvalue must A have?
5. What do the eigenvectors of a covariance matrix represent (in PCA)?

Answers

Reveal answers

1. Three over the complex numbers, counting algebraic multiplicity. Not all three must be real.
2. Yes — det(A−λI) = det((A−λI)ᵀ) = det(Aᵀ−λI), so A and Aᵀ share eigenvalues.
3. False (only if A has n linearly independent eigenvectors).
4. λ = 0 (since det(A) = product of eigenvalues).
5. The principal components / axes of greatest data variance.

Quick revision box

• Eigenvalue/eigenvector: Av = λv; solve det(A−λI)=0 for λ, then (A−λI)v=0.
• Characteristic polynomial is degree n ⇒ n eigenvalues (with multiplicity).
• det = product of λ; trace = sum of λ.
• Diagonalisation A = P D P⁻¹; powers Aᵏ = P Dᵏ P⁻¹ (if diagonalisable).
• Geometric: eigenvectors = pure-stretc. directions; eigenvalues = stretch factors.
• Applications: stability, PCA (eigenvectors of covariance), PageRank (λ=1), vibrations.
• Similarity B=P⁻¹AP ⇒ same eigenvalues; diagonalisation is similarity to a diagonal matrix.

Practice ladder

1. Easy - Recall: Define the module's central idea in one or two sentences.
2. Easy - Recognize: Identify the correct method for a small example and explain why it fits.
3. Medium - Apply: Work through one representative problem without copying the example.
4. Medium - Compare: Contrast two methods or concepts from the module.
5. Hard - Integrate: Solve a university-style scenario and justify every major step.

Reveal self-evaluation guide

A complete response uses correct terminology, shows intermediate steps, connects the result to the scenario, and states one assumption or limitation.

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