Eigenvalues and Eigenvectors

Eigenvalues are the "natural numbers" attached to a matrix — they reveal how the matrix stretches, rotates and twists space along its preferred directions. For GATE CSE, a small set of properties lets you read off most answers in seconds, without ever solving the characteristic polynomial.

Definition: For a square matrix A of size n × n, a non-zero vector v is an eigenvector and the scalar λ is its eigenvalue if Av = λv. The action of A on v is reduced to a simple scaling by λ — no rotation, no shear, just a stretch (and a flip if λ is negative).

Definition: The characteristic equation of A is det(A − λI) = 0. Its roots are exactly the eigenvalues of A; the equation packages all eigenvalues into one polynomial of degree n.

How the characteristic equation encodes everything

If you expand det(A − λI), you get a polynomial of degree n in λ called the characteristic polynomial. Its coefficients carry information about every algebraic invariant of the matrix. Two of these invariants matter most for GATE-style MCQs and NATs:

• The sum of the eigenvalues equals the trace of A (the sum of its diagonal entries): λ₁ + λ₂ + … + λₙ = tr(A).
• The product of the eigenvalues equals the determinant of A: λ₁ · λ₂ · … · λₙ = det(A).

These are not accidents. They are Vieta's formulas applied to the characteristic polynomial. Whenever a question gives you tr(A) and det(A) for a 2×2 matrix, you already have the eigenvalues, even without writing the polynomial.

Triangular and diagonal matrices: read eigenvalues straight off

For an upper-triangular, lower-triangular or diagonal matrix, the eigenvalues are simply the diagonal entries. This is because det(A − λI) for a triangular matrix is the product of its diagonal entries (a₁₁ − λ)(a₂₂ − λ)…(aₙₙ − λ), so its roots are a₁₁, a₂₂, …, aₙₙ. GATE often hides a triangular matrix inside a question — spot it and skip the algebra.

How eigenvalues transform under common operations

If λ is an eigenvalue of A with eigenvector v, then a function of A often has the same eigenvector with a transformed eigenvalue:

• A^k: eigenvalues are λ^k. Proof: A²v = A(Av) = A(λv) = λ(Av) = λ²v.
• A⁻¹ (only if A is invertible): eigenvalues are 1/λ. Proof: from Av = λv, multiply both sides by A⁻¹/λ to get A⁻¹v = (1/λ)v.
• A^T (transpose): same eigenvalues as A (though not the same eigenvectors in general). Because det(A^T − λI) = det((A − λI)^T) = det(A − λI).
• A + cI: eigenvalues shift by c, giving λ + c. The eigenvectors stay the same.
• kA (scalar multiple): eigenvalues are kλ.

Why it matters: many GATE problems give you a polynomial expression in A — say, A² − 3A + 2I — and ask for its eigenvalues. You don't have to compute the whole matrix. Just substitute each eigenvalue λ of A: the eigenvalues of A² − 3A + 2I are simply λ² − 3λ + 2. This spectral mapping rule is the single most useful trick in this topic.

A worked example using only the properties

Question: For A = [[4, 1], [2, 3]], find the eigenvalues of A² − 5A + 6I.
Solution:
Step 1: tr(A) = 4 + 3 = 7 and det(A) = 4·3 − 1·2 = 10. So sum of eigenvalues = 7 and product = 10.
Step 2: Solve λ² − 7λ + 10 = 0 to get λ = 2 and λ = 5.
Step 3: For any polynomial p, the eigenvalues of p(A) are p(λ). Here p(λ) = λ² − 5λ + 6.
Step 4: p(2) = 4 − 10 + 6 = 0; p(5) = 25 − 25 + 6 = 6.
Conclusion: Eigenvalues of A² − 5A + 6I are 0 and 6. Because 0 is one of them, the matrix is singular.

Singular matrices, symmetry and a few elegant facts

A matrix is singular (non-invertible) if and only if 0 is an eigenvalue of A — equivalent to det(A) = 0 — because the product of eigenvalues is the determinant. This single line answers many "is A invertible?" questions instantly.

For a real symmetric matrix (A = A^T), two famous facts hold:

1. All eigenvalues are real (no imaginary parts at all).
2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

These together drive the spectral theorem, which underlies PCA, Google's PageRank power-iteration, image compression by SVD, and the analysis of graph Laplacians.

The Cayley–Hamilton theorem

Cayley–Hamilton says: every square matrix satisfies its own characteristic equation. If p(λ) = det(λI − A) is the characteristic polynomial, then p(A) = 0 — the zero matrix. This lets you express A^n (for n ≥ size of A) as a polynomial in A of degree less than n, which is the GATE shortcut for computing high powers and inverses of matrices.

Real-world example: When Google's original PageRank algorithm ran, it was computing the dominant eigenvector of a 25-billion-row matrix representing the web. The dominant eigenvalue was exactly 1 (since the matrix is row-stochastic), and the corresponding eigenvector gave every web page its rank. Eigenvalues are the heartbeat of search engines, neural-network stability analysis, and quantum mechanics.

Common misconception: "Eigenvalues of A and A^T are different because the matrices are different." Wrong — they are equal, because det(A − λI) = det((A − λI)^T) = det(A^T − λI). The eigenvectors, however, are not generally the same.

Common misconception: "If A has eigenvalues 2 and 3, then A + I has eigenvalues 2 and 3 too." Wrong — shifting A by cI shifts every eigenvalue by c, so A + I has eigenvalues 3 and 4. You can verify by substituting (A + I)v = Av + v = λv + v = (λ + 1)v.

Common misconception: "An n×n matrix can have fewer than n eigenvalues." It always has exactly n eigenvalues counted with algebraic multiplicity (over complex numbers). It may have fewer distinct eigenvalues, but the total count from the characteristic polynomial is always n.

A is diagonalizable iff it has n linearly independent eigenvectors; then A = PDP^-1 where D is diagonal with eigenvalues and P's columns are eigenvectors. This makes A^k = PD^kP^-1 trivial. Distinct eigenvalues guarantee diagonalizability (sufficient, not necessary). Algebraic multiplicity ≥ geometric multiplicity; defective matrices (geometric < algebraic) are NOT diagonalizable. Cayley-Hamilton theorem: if characteristic polynomial is p(λ), then p(A) = 0; this lets you express A^n and A^-1 in terms of lower powers. Symmetric matrices are always diagonalizable by an orthogonal matrix. Use these to compute high powers of matrices efficiently in exams.

Find eigenvalues of A = [[4,1],[2,3]]. Characteristic equation: det(A − λI) = (4−λ)(3−λ) − (1)(2) = 0. Expand: 12 − 4λ − 3λ + λ^2 − 2 = λ^2 − 7λ + 10 = 0. Factor: (λ − 5)(λ − 2) = 0, so λ = 5, 2. Verify: trace = 4 + 3 = 7 = 5 + 2 ✓; det = 4·3 − 1·2 = 10 = 5 × 2 ✓. Shortcut for 2×2: λ^2 − (trace)λ + det = 0. So you can write λ^2 − 7λ + 10 = 0 directly without expanding. Always cross-check with trace and determinant.

Cover the answer, recall, then check. 12 cards on Cayley–Hamilton, diagonalization, and eigenvalue transformations — the higher-mark GATE angle.

Q1. State the Cayley–Hamilton theorem.
A1. Every square matrix satisfies its own characteristic equation: if p(λ) = det(A − λI), then p(A) = 0 (the zero matrix).

Q2. How is Cayley–Hamilton used to compute A⁻¹?
A2. From p(A) = 0, isolate the identity term and express A⁻¹ as a polynomial in A (valid when the constant term, ±det(A), is nonzero).

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

Q4. What are the eigenvalues of adj(A) in terms of λ and det(A)?
A4. det(A)/λ for each eigenvalue λ of A.

Q5. State the condition for A (n×n) to be diagonalizable.
A5. A has n linearly independent eigenvectors ⇔ geometric multiplicity = algebraic multiplicity for every eigenvalue.

Q6. If A = PDP⁻¹ (D diagonal), what is Aᵏ?
A6. Aᵏ = P Dᵏ P⁻¹, where Dᵏ just raises each diagonal eigenvalue to the k-th power — the reason diagonalization makes powers cheap.

Q7. Do distinct eigenvalues guarantee diagonalizability?
A7. Yes — if an n×n matrix has n distinct eigenvalues it is automatically diagonalizable (eigenvectors are independent).

Q8. What do similar matrices (B = P⁻¹AP) share?
A8. Same eigenvalues, characteristic polynomial, trace, determinant, and rank. (Eigenvectors transform by P.)

Q9. What can you say about eigenvalues/eigenvectors of a real symmetric matrix?
A9. All eigenvalues are real, and eigenvectors for distinct eigenvalues are orthogonal; it is always diagonalizable by an orthogonal matrix.

Q10. Eigenvalues of skew-symmetric and orthogonal matrices?
A10. Skew-symmetric: 0 or purely imaginary. Orthogonal: |λ| = 1 (may be complex on the unit circle).

Q11. How do you find A¹⁰⁰ for a diagonalizable A efficiently?
A11. Diagonalize A = PDP⁻¹, then A¹⁰⁰ = P D¹⁰⁰ P⁻¹ — raise only the diagonal eigenvalues to the 100th power.

Q12. What does positive definite mean in eigenvalue terms?
A12. A symmetric matrix is positive definite ⇔ all eigenvalues > 0 (positive semidefinite ⇔ all ≥ 0).

Beyond simply computing eigenvalues, GATE CSE tests what you can do with them: Cayley–Hamilton to compute inverses/powers, diagonalization to raise a matrix to a high power, and the eigenvalue transformation rules. These carry 1–2 marks and reward pattern recognition over grinding.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic equation p(A) = 0. This lets you (a) express high powers Aⁿ, Aⁿ⁺¹ … in terms of lower powers, and (b) write A⁻¹ as a polynomial in A. It converts "compute A⁵" into a couple of substitutions.

Diagonalization

If A has n independent eigenvectors, then A = PDP⁻¹ with D diagonal (eigenvalues) and P their eigenvectors. Then Aᵏ = P Dᵏ P⁻¹ — powers become trivial. A is diagonalizable ⇔ GM = AM for every eigenvalue; n distinct eigenvalues is a sufficient condition.

Eigenvalue transformation rules

Matrix	Its eigenvalues
Aᵏ	λᵏ
A⁻¹	1/λ
A + cI	λ + c
adj(A)	det(A)/λ
kA	kλ

Eigenvectors stay the same for all of these.

Exam Tricks & Tips

• 🎯 Cayley–Hamilton for powers: to find A³ or A⁻¹ of a 2×2, substitute A into its own characteristic equation instead of multiplying.
• 🎯 Diagonalize before taking high powers — A⁵⁰ via P D⁵⁰ P⁻¹ is far faster than repeated multiplication.
• 🎯 Distinct eigenvalues ⇒ diagonalizable, so don't bother checking multiplicities in that case.
• 🎯 Symmetric ⇒ real eigenvalues + orthogonal eigenvectors + always diagonalizable — a huge shortcut.
• 🎯 Similar matrices share trace, det, eigenvalues, rank — use these invariants to reject options.
• ❌ Common mistake: assuming every matrix is diagonalizable. A defective matrix (GM < AM, e.g. a Jordan block) is not, and its powers cannot use the PDᵏP⁻¹ shortcut.

Expected exam pattern

Apply Cayley–Hamilton to compute an inverse or a specified power; determine whether a given matrix is diagonalizable; find eigenvalues of A⁻¹, Aᵀ, adj(A), or A + cI given those of A; or use symmetric-matrix properties.

Quick recap

Cayley–Hamilton: p(A)=0 → powers and inverse as polynomials in A. Diagonalizable ⇔ n independent eigenvectors (GM=AM); then Aᵏ=PDᵏP⁻¹. Transform rules: Aᵏ→λᵏ, A⁻¹→1/λ, A+cI→λ+c, adj A→det(A)/λ, eigenvectors unchanged. Symmetric ⇒ real, orthogonal, diagonalizable.