Eigenvalues, eigenvectors and diagonalisation

Learning objectives

Form the characteristic equation det(A − λI) = 0 for a 2×2 or 3×3 matrix and solve it to obtain the eigenvalues, for matrices whose eigenvalues are real and distinct (the only case the syllabus sets)

Find an eigenvector for each eigenvalue by solving the homogeneous system (A − λI)x = 0, and state the general eigenvector as a scalar multiple of a chosen non-zero solution

Construct a matrix P whose columns are linearly independent eigenvectors and the diagonal matrix D of the corresponding eigenvalues, and verify that A = PDP⁻¹, knowing that distinct eigenvalues always give independent eigenvectors, so that P is invertible

Evaluate a matrix power using Aⁿ = PDⁿP⁻¹, where Dⁿ is obtained by raising each diagonal entry to the power n, and use this to solve a problem about a repeated transformation

State the Cayley–Hamilton theorem, that every square matrix satisfies its own characteristic equation, and use it to express Aⁿ or A⁻¹ as a linear combination of lower powers of A and the identity

Relate the real eigenvectors of a 2×2 matrix to the invariant lines through the origin of the transformation it represents, and interpret an eigenvalue as the scale factor along such a line