Eigenvalues and Eigenvectors

Eigenvalues/eigenvectors are a GATE EC favourite (nearly every year, 1–2 marks) and underpin state-space, stability, and system-mode analysis. Av = λv means v keeps its direction under A, scaled by λ.

Key results

• Characteristic equation: det(A − λI) = 0.
• Sum of eigenvalues = trace(A); product of eigenvalues = det(A).
• Eigenvalues of A⁻¹ are 1/λ; of Aᵏ are λᵏ; of A+kI are λ+k; A and Aᵀ share eigenvalues.
• Triangular/diagonal matrix: eigenvalues = diagonal entries.
• Symmetric (real) → real eigenvalues, orthogonal eigenvectors. Skew-symmetric → purely imaginary/zero. Orthogonal → |λ| = 1.
• Cayley–Hamilton: every matrix satisfies its own characteristic equation (use to compute Aⁿ, A⁻¹).

Operation on A	Effect on eigenvalue λ	Eigenvector
A⁻¹	1/λ	same
Aᵏ	λᵏ	same
A + kI	λ + k	same
kA	kλ	same
Aᵀ	λ (same set)	may differ

Exam Tricks & Tips

• 🎯 Don't expand the characteristic polynomial if trace and det pin down both eigenvalues of a 2×2 (λ₁+λ₂ = trace, λ₁λ₂ = det).
• 🎯 Idempotent matrix (A² = A) → eigenvalues 0 or 1; nilpotent (Aᵏ = 0) → all eigenvalues 0.
• 🎯 Cayley–Hamilton turns high powers into low-order polynomials — huge time saver for Aⁿ.
• 🎯 Repeated eigenvalue may lack a full set of independent eigenvectors (defective → not diagonalizable).
• 🎯 For a projection/rotation matrix, reason geometrically instead of computing.
• ❌ Common mistake: adding k to eigenvalues of kA — scaling A multiplies eigenvalues by k; adding kI adds k. Keep them separate.

Expected exam pattern

1-mark: eigenvalue of a 2×2/3×3, or of A⁻¹/A²/A+I given A's eigenvalues. 2-mark: eigenvector for a given λ, or apply Cayley–Hamilton to find A⁻¹ or a high power.

Quick recap

Solve det(A−λI)=0. Sum λ = trace, product λ = det. λ transforms predictably: 1/λ for inverse, λᵏ for power, λ+k for A+kI. Symmetric → real λ; Cayley–Hamilton reduces powers.

Cover the answer, recall, then check. 12 cards on eigenvalues and eigenvectors for GATE EC.

Q1. What equation gives the eigenvalues of A?
A1. The characteristic equation det(A − λI) = 0.

Q2. Relate sum and product of eigenvalues to A.
A2. Sum of eigenvalues = trace(A); product of eigenvalues = det(A).

Q3. If λ is an eigenvalue of A, what are the eigenvalues of A⁻¹, Aᵏ, and A + kI?
A3. 1/λ, λᵏ, and λ + k respectively — with the same eigenvector.

Q4. What are the eigenvalues of a triangular matrix?
A4. Its diagonal entries.

Q5. Eigenvalue nature: symmetric vs skew-symmetric vs orthogonal (real matrices)?
A5. Symmetric → real; skew-symmetric → purely imaginary or zero; orthogonal → magnitude 1.

Q6. State the Cayley–Hamilton theorem and one use.
A6. Every square matrix satisfies its own characteristic equation. Use it to express Aⁿ or A⁻¹ as a low-order polynomial in A.

Q7. Eigenvalues of an idempotent matrix (A² = A)?
A7. Only 0 and 1.

Q8. Eigenvalues of a nilpotent matrix (Aᵏ = 0)?
A8. All eigenvalues are 0.

Q9. Do A and Aᵀ have the same eigenvalues?
A9. Yes — same characteristic polynomial, so same eigenvalues (eigenvectors may differ).

Q10. When is a matrix diagonalizable?
A10. When it has n linearly independent eigenvectors (guaranteed if all eigenvalues are distinct, or if it is real symmetric).

Q11. For a 2×2 with trace 5 and det 6, find the eigenvalues.
A11. Roots of λ² − 5λ + 6 = 0 → λ = 2 and λ = 3.

Q12. What is a "defective" matrix?
A12. One with a repeated eigenvalue whose eigenvectors are fewer than its algebraic multiplicity → not diagonalizable.

Eigenvalues reveal the "natural axes" of a matrix — directions that a linear transformation only stretches, never rotates. They decide the stability of state-space systems, the modes of vibrating structures, and the diagonalisation that makes matrix powers trivial. GATE treats them as near-guaranteed marks, usually as a fast computation exploiting trace and determinant.

Core concept: For A·v = λ·v, λ is an eigenvalue and v its eigenvector: v points along an invariant direction that A merely scales by λ.

The theory

Beginner — the characteristic equation

Eigenvalues solve det(A − λI) = 0, a degree-n polynomial. For each λ, the eigenvector is found from (A − λI)v = 0 (the null space). Two invariants make computation fast:

• Sum of eigenvalues = trace(A) (sum of diagonal entries).
• Product of eigenvalues = det(A).
  For a 2×2 matrix these two facts alone often give both eigenvalues by inspection.

Intermediate — properties

• Eigenvalues of a triangular/diagonal matrix are its diagonal entries.
• Symmetric real matrix → all eigenvalues real, eigenvectors orthogonal.
• Skew-symmetric → eigenvalues purely imaginary or zero.
• Orthogonal → eigenvalues have magnitude 1.
• Eigenvalues of Aⁿ are λⁿ; of A^(−1) are 1/λ; of A + kI are λ + k. Eigenvectors are unchanged by these operations.
• A singular matrix has at least one zero eigenvalue (det = 0 = Πλ).

Advanced — diagonalisation and Cayley-Hamilton

If A has n linearly independent eigenvectors, A = P·D·P^(−1) with D = diag(λi) and P their eigenvectors. Then Aᵏ = P·Dᵏ·P^(−1) — powers become trivial. Repeated eigenvalues may fail to give enough independent eigenvectors (defective matrix), needing Jordan form.
Cayley-Hamilton theorem: A satisfies its own characteristic polynomial, e.g. A² − (trace)A + det·I = 0 for 2×2 — a slick way to compute A^(−1) or reduce high powers.

Worked example

Find the eigenvalues of A = [4 1; 2 3].
trace = 4 + 3 = 7 = λ1 + λ2; det = 4·3 − 1·2 = 10 = λ1·λ2.
So λ² − 7λ + 10 = 0 → (λ−5)(λ−2) = 0 → λ = 5, 2.
Eigenvalues of A^(−1) would be 1/5 and 1/2; of A + 3I would be 8 and 5 — no fresh computation needed.

GATE relevance

Almost every GATE has an eigenvalue item: find eigenvalues/eigenvectors, use trace/det shortcuts, eigenvalues of A²/A^(−1)/A+kI, diagonalisability, or apply Cayley-Hamilton. Also underpins state-space stability (eigenvalues = poles) in Control Systems.

Exam tricks and shortcuts

• Use λ1+λ2 = trace, λ1·λ2 = det for 2×2 — usually no polynomial needed.
• Eigenvalues of A^(−1) = 1/λ, of Aⁿ = λⁿ, of A+kI = λ+k (same eigenvectors).
• Triangular matrix ⇒ eigenvalues sit on the diagonal.
• Mnemonic: "Trace sums, Det multiplies" the eigenvalues.

Key formulas

• Characteristic equation: |A − λI| = 0.
• Eigenvector: (A − λI)x = 0, x ≠ 0.
• Sum of eigenvalues = trace(A); product = |A|.
• Properties: eigenvalues of Aⁿ are λⁿ; of A⁻¹ are 1/λ; of kA are kλ.
• Symmetric matrix: real eigenvalues, orthogonal eigenvectors.
• Diagonalisation: A = PDP⁻¹ (D = diag of eigenvalues).

Eigenvalues characterise a matrix's scaling directions; symmetric matrices are always diagonalisable.

Worked Example

Problem: For A = [[4, 1], [2, 3]], find the eigenvalues and the eigenvector associated with the larger eigenvalue.

Solution:
The characteristic equation is det(A − λI) = 0:
|[[4−λ, 1], [2, 3−λ]]| = (4−λ)(3−λ) − (1)(2) = 0.
Expand: λ² − 7λ + 12 − 2 = λ² − 7λ + 10 = 0.
Factor: (λ − 5)(λ − 2) = 0, so λ₁ = 5 and λ₂ = 2.

Sanity checks:
trace(A) = 4 + 3 = 7 = λ₁ + λ₂ ✓, and det(A) = 12 − 2 = 10 = λ₁λ₂ ✓.

Eigenvector for λ = 5: solve (A − 5I)v = 0.
A − 5I = [[−1, 1], [2, −2]].
Row 1 gives −x + y = 0 ⇒ y = x. (Row 2 is just −2 times row 1, so it is redundant.)
Choosing x = 1 gives the eigenvector v = [1, 1]ᵀ.
Verify: A[1, 1]ᵀ = [4+1, 2+3]ᵀ = [5, 5]ᵀ = 5[1, 1]ᵀ ✓.

Answer: Eigenvalues λ = 5 and λ = 2; eigenvector for λ = 5 is [1, 1]ᵀ (any nonzero scalar multiple).