MTH2021 · Linear Algebra with Applications
Eigenvalues, Eigenvectors & Diagonalisation
Weeks 10–11 solve Ax = λx: eigenvalues from the characteristic equation det(λI − A) = 0, eigenspaces as kernels of λI − A, algebraic vs geometric multiplicity, and diagonalisation A = PDP⁻¹ when A has n independent eigenvectors. Diagonalisation makes powers Aᵏ = PDᵏP⁻¹ easy and drives applications to Markov chains and difference/differential equations. This is a cornerstone of the comprehensive 50% final.
What this chapter covers
- 01Eigenvector x ≠ 0 with Ax = λx; scalar multiples cx are also eigenvectors
- 02Characteristic equation det(λI − A) = 0; the characteristic polynomial is monic of degree n, so ≤ n distinct eigenvalues
- 03Triangular matrices have their diagonal entries as eigenvalues; A invertible ⇔ 0 is not an eigenvalue
- 04Eigenspace = ker(λI − A); λ eigenvalue of A ⇒ λᵏ eigenvalue of Aᵏ (same eigenvector)
- 05Algebraic multiplicity (power in the characteristic polynomial) vs geometric multiplicity (dim of eigenspace); geometric ≤ algebraic
- 06Similar matrices B = P⁻¹AP share determinant, characteristic polynomial and geometric multiplicities
- 07Diagonalisable ⇔ n linearly independent eigenvectors ⇔ each eigenvalue's geometric = algebraic multiplicity; n distinct eigenvalues ⇒ diagonalisable
- 08Diagonalisation A = PDP⁻¹ and its use: Aᵏ = PDᵏP⁻¹, Markov chains, difference/differential equations
Diagonalising a 2×2 matrix
- +1Characteristic equation: det(A − λI) = det[[2−λ, 1], [1, 2−λ]] = (2−λ)² − 1 = λ² − 4λ + 3 = (λ − 1)(λ − 3). So the eigenvalues are λ = 1 and λ = 3.
- +1Eigenvector for λ = 1: (A − I) = [[1, 1], [1, 1]], giving x₁ + x₂ = 0, so an eigenvector is (1, −1).
- +1Eigenvector for λ = 3: (A − 3I) = [[−1, 1], [1, −1]], giving −x₁ + x₂ = 0, so an eigenvector is (1, 1).
- +1Assemble P from the eigenvectors as columns and D from the matching eigenvalues: P = [[1, 1], [−1, 1]], D = [[1, 0], [0, 3]]. (Order must match: column 1 of P pairs with D₁₁ = 1.)
- +1Verify A = PDP⁻¹. Since the eigenvalues are distinct, the eigenvectors are independent, so P is invertible and A is diagonalisable. Check AP = PD: A·(1,−1)ᵀ = (1,−1)ᵀ = 1·(1,−1)ᵀ and A·(1,1)ᵀ = (3,3)ᵀ = 3·(1,1)ᵀ ✓. A useful consequence: Aᵏ = PDᵏP⁻¹.
Key terms
- Eigenvalue / eigenvector
- A scalar λ and nonzero vector x with Ax = λx; the eigenvalues are the roots of the characteristic equation det(λI − A) = 0.
- Characteristic polynomial
- det(λI − A), a monic polynomial of degree n whose roots are the eigenvalues; A has at most n distinct eigenvalues.
- Eigenspace
- ker(λI − A), the subspace of all eigenvectors for a given eigenvalue λ (together with 0); its dimension is the geometric multiplicity.
- Algebraic vs geometric multiplicity
- Algebraic = the multiplicity of λ as a root of the characteristic polynomial; geometric = dim of the eigenspace. Always geometric ≤ algebraic.
- Similar matrices
- B = P⁻¹AP for some invertible P; similar matrices share determinant, characteristic polynomial, eigenvalues and geometric multiplicities.
- Diagonalisation
- A = PDP⁻¹ with D diagonal and P's columns the eigenvectors; possible iff A has n linearly independent eigenvectors, and then Aᵏ = PDᵏP⁻¹.
Eigenvalues, Eigenvectors & Diagonalisation FAQ
How do I find eigenvalues and eigenvectors?
Solve the characteristic equation det(λI − A) = 0 for the eigenvalues, then for each λ find the eigenspace by computing ker(λI − A) — the nonzero solutions of (A − λI)x = 0. Scalar multiples of an eigenvector are again eigenvectors, so you report a basis of each eigenspace.
When is a matrix diagonalisable?
Exactly when it has n linearly independent eigenvectors, equivalently when every eigenvalue's geometric multiplicity equals its algebraic multiplicity. A quick sufficient condition: n distinct eigenvalues force independence, so the matrix is diagonalisable (the converse is false — repeated eigenvalues can still diagonalise).
What is the difference between algebraic and geometric multiplicity?
Algebraic multiplicity is how many times λ is a root of the characteristic polynomial; geometric multiplicity is the dimension of its eigenspace ker(λI − A). Geometric is always at most algebraic, and diagonalisability is precisely the case where they are equal for every eigenvalue. A gap means the matrix is defective.
Why does diagonalisation make powers easy?
If A = PDP⁻¹ then Aᵏ = PDᵏP⁻¹, and DᵏT is just the diagonal entries raised to the k. So computing A¹⁰⁰, iterating a Markov chain, or solving a linear difference/differential system reduces to powering scalars — the main practical payoff of the whole chapter.
Where is this examined?
Eigenvalues and diagonalisation (Weeks 10–11) fall after Test 2's window, so they are assessed in their weekly quizzes and, above all, in the comprehensive 50% final. Expect a full diagonalisation plus an application (a power of A or a defective-matrix multiplicity argument).
Exam move
Practise the pipeline end to end: characteristic polynomial → eigenvalues → eigenspaces ker(λI − A) → assemble P and D with columns and diagonal aligned. Always check AP = PD rather than inverting P, and be ready to argue diagonalisability via geometric vs algebraic multiplicity, including a defective (non-diagonalisable) example. Rehearse the Aᵏ = PDᵏP⁻¹ application. This is a cornerstone of the 50% final. When your eigenvectors don't satisfy Ax = λx, ask Sia to recheck the (A − λI)x = 0 solve.
Working through Eigenvalues, Eigenvectors & Diagonalisation in MTH2021? Sia is AskSia’s AI Mathematics tutor — ask any MTH2021 Eigenvalues, Eigenvectors & Diagonalisation question and get a clear, step-by-step explanation grounded in how MTH2021 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
