Eigenvalues

Eigenvalues

1. Definition

For a square matrix A of order n, a scalar λ is an eigenvalue (characteristic value / latent root) of A if there exists a non-zero column vector X such that

X is then the corresponding eigenvector (next topic). Rearranging gives (A − λI)X = 0 — a homogeneous system — and such a system has a non-trivial solution only when its coefficient matrix is singular:

This is the characteristic equation; det(A − λI) expanded is the characteristic polynomial of degree n. Its n roots (counting multiplicity) are the n eigenvalues.

2. Recipe for Finding Eigenvalues

1. Form A − λI: subtract λ from every diagonal entry (off-diagonal entries are untouched).
2. Expand det(A − λI) = 0 → a quadratic for 2 × 2, a cubic for 3 × 3.
3. Solve. For a cubic, try the easy roots first (0, ±1, ±2, or a value that makes two rows equal); then divide out the factor to get the rest.
4. Always verify with: λ₁ + λ₂ + … + λₙ = trace A and λ₁λ₂…λₙ = det A.

3. Properties of Eigenvalues (high-value exam list)

Property	Statement
Number	An n × n matrix has exactly n eigenvalues, counting multiplicity (real or complex)
Sum	λ₁ + λ₂ + … + λₙ = trace A (sum of the diagonal entries)
Product	λ₁ λ₂ … λₙ = det A
Triangular / diagonal	Eigenvalues are simply the diagonal entries
Zero eigenvalue	λ = 0 is an eigenvalue ⇔ A is singular (det A = 0)
Transpose	A and Aᵀ have the same eigenvalues
Powers	Eigenvalues of Aᵐ are λᵐ
Scalar multiple	Eigenvalues of kA are kλ
Inverse	If A is non-singular, the eigenvalues of A⁻¹ are 1/λ
Similar matrices	Similar matrices have identical eigenvalues (not the same eigenvectors)
Symmetric / Hermitian	All eigenvalues are real
Real matrix	Complex eigenvalues occur in conjugate pairs

        [ 4  1 ]
   A =  [ 2  3 ]

   A - λI = [ 4-λ   1  ]   ⇒  det = (4-λ)(3-λ) - (1)(2) = λ² - 7λ + 10 = 0
            [  2   3-λ ]

   (λ - 5)(λ - 2) = 0   ⇒   λ = 5 , λ = 2
   Checks:  5 + 2 = 7 = trace(4+3) ✓      5 × 2 = 10 = det ✓

7. Typical Exam Variations

1. Find all eigenvalues of a 2 × 2 or 3 × 3 matrix — the basic question.
2. Given two eigenvalues, find the third from sum = trace or product = det (no expansion needed) — very common.
3. Find k such that a stated number is an eigenvalue: substitute λ into det(A − λI) = 0 and solve for k.
4. Prove a property — e.g. that a singular matrix has 0 as an eigenvalue, that A and Aᵀ have the same eigenvalues, that similar matrices share eigenvalues.
5. Symmetric matrices: state (or prove) that all eigenvalues are real.
6. Word problem: eigenvalues of A⁻¹, A², A + 2I etc. given the eigenvalues of A.

8. Exam Points

• Define an eigenvalue and obtain the characteristic equation det(A − λI) = 0 — 5 marks, almost certain.
• Find the eigenvalues of a given 2 × 2 and 3 × 3 matrix — 5-7 marks, very frequent.
• State and apply the properties (sum = trace, product = det, diagonal entries, A⁻¹, Aᵐ, kA, similarity) — 5 marks, frequent.
• Find the missing eigenvalue using trace/determinant — 3-5 marks.
• Prove that a singular matrix has a zero eigenvalue, and conversely — 3 marks.
• Show that a real symmetric matrix has real eigenvalues — 5 marks.
• Find k such that a given number is an eigenvalue of A — 5 marks.