Topic 4 of 6 · Form 6
Further Pure 2: complex numbers and matrices (eigenvalues)
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ; expressing cos nθ and sin nθ in powers of cos θ and sin θ; expressing powers (e.g. sin⁵θ) in multiple angles using z ± 1/z; summing series with complex exponentials.
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de Moivre's theorem
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ; expressing cos nθ and sin nθ in powers of cos θ and sin θ; expressing powers (e.g. sin⁵θ) in multiple angles using z ± 1/z; summing series with complex exponentials.
Worked example: Show that cos 4θ = 8cos⁴θ − 8cos²θ + 1.
Real part of (c + is)⁴ = c⁴ − 6c²s² + s⁴; with s² = 1 − c²: c⁴ − 6c² + 6c⁴ + 1 − 2c² + c⁴ = 8c⁴ − 8c² + 1
nth roots
Roots of zⁿ = w, roots of unity, geometric interpretation as regular polygons.
Eigenvalues and eigenvectors
Characteristic equation det(A − λI) = 0; eigenvectors; diagonalisation A = PDP⁻¹ (and for symmetric matrices with orthogonal P); powers of matrices; Cayley–Hamilton theorem.
Worked example: A = (5 −2; 1 2). Find the eigenvalues.
(5 − λ)(2 − λ) + 2 = λ² − 7λ + 12 = 0 → λ = 3, 4
Check yourself
- Find the cube roots of 8i in exponential form.
- What is the trace of (5 −2; 1 2)?
- State the Cayley–Hamilton theorem.
Show answers
- 2e^(iπ/6), 2e^(i5π/6), 2e^(−iπ/2)
- 7
- Every square matrix satisfies its own characteristic equation.
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