Eigenvalues & Eigenvectors – Engineering Maths for GATE, ESE, AE/JE

Eigenvalues and Eigenvectors are among the most important topics in Linear Algebra under Engineering Mathematics. This topic is frequently asked in GATE, ESE, AE/JE exams and is essential for advanced concepts like diagonalization.

Basic Concept

For a square matrix A, if there exists a scalar λ (lambda) and a non-zero vector X such that:

AX = λX

Then:

• λ is called an Eigenvalue
• X is called the corresponding Eigenvector

Characteristic Equation

Eigenvalues are obtained by solving:

|A − λI| = 0

This is called the characteristic equation.

Steps to Find Eigenvalues

1. Form matrix (A − λI)
2. Find determinant |A − λI|
3. Solve the equation = 0

Steps to Find Eigenvectors

1. Substitute λ in (A − λI)X = 0
2. Solve system of equations
3. Find non-zero vector X

Important Properties

• Sum of eigenvalues = Trace of matrix
• Product of eigenvalues = Determinant of matrix
• Eigenvalues of triangular matrix = diagonal elements
• If A is singular → one eigenvalue = 0
• Eigenvalues of A and A^T are same

Special Cases (VERY IMPORTANT)

• Symmetric Matrix: Eigenvalues are real
• Skew-Symmetric Matrix: Eigenvalues are zero or purely imaginary
• Orthogonal Matrix: |λ| = 1
• Idempotent Matrix: λ = 0 or 1

Solved Examples

Example 1

Find eigenvalues of:

|A − λI| =

(2−λ)(3−λ) = 0

Eigenvalues: λ = 2, 3

Example 2

Find eigenvalues of:

|A − λI| =

(4−λ)(3−λ) − 2 = 0
λ² − 7λ + 10 = 0

Eigenvalues: λ = 5, 2

Example 3 (Eigenvector)

Find eigenvector corresponding to λ = 2 for:

(A − 2I)X = 0

⇒ y = 0, x = free

Eigenvector: (1, 0)

Example 4

If trace = 6 and determinant = 8 for 2×2 matrix, find eigenvalues.

λ₁ + λ₂ = 6
λ₁λ₂ = 8

λ² − 6λ + 8 = 0

Eigenvalues: 4, 2

Shortcut Tricks

• Use trace & determinant directly (2×2 matrix)
• Diagonal matrix → eigenvalues = diagonal elements
• Check zero determinant → λ = 0
• Avoid full expansion when possible

Questions

Q1. Sum of eigenvalues equals?

Answer: Trace

Q2. Product of eigenvalues equals?

Answer: Determinant

Q3. Eigenvalues of identity matrix?

Answer: All 1

Exam Tips

• Memorize trace & determinant relation
• Use shortcuts for 2×2 matrices
• Practice PYQs
• Avoid calculation mistakes in determinant

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