Diagonalization of Matrix – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

Diagonalization of a matrix is an important concept in Linear Algebra. It is widely used in simplifying matrix powers and is frequently asked in GATE, ESE, AE/JE exams.

Basic Concept

A square matrix A is said to be diagonalizable if it can be written as:

A = P D P⁻¹

Where:

• D = Diagonal matrix (contains eigenvalues)
• P = Matrix of eigenvectors
• P⁻¹ = Inverse of P

Condition for Diagonalization

• Matrix must have n linearly independent eigenvectors
• Distinct eigenvalues → always diagonalizable
• If eigenvalues are repeated → check eigenvectors

Steps to Diagonalize a Matrix

1. Find eigenvalues of matrix A
2. Find corresponding eigenvectors
3. Form matrix P using eigenvectors as columns
4. Form diagonal matrix D using eigenvalues
5. Verify: A = P D P⁻¹

Diagonal Matrix Form

If eigenvalues are λ₁, λ₂, then:

Important Properties

• If A = P D P⁻¹, then Aⁿ = P Dⁿ P⁻¹
• Dⁿ is easy to compute (just raise diagonal elements)
• Trace = sum of eigenvalues
• Determinant = product of eigenvalues

Solved Examples

Example 1

Diagonalize:

Eigenvalues = 2, 3
Matrix is already diagonal

D = same matrix, P = Identity

Example 2

Diagonalize:

Eigenvalues = 5, 2

Eigenvectors:

• For λ = 5 → (1,1)
• For λ = 2 → (1,-2)

P =

D =

Thus: A = P D P⁻¹

Example 3

If A is diagonalizable and eigenvalues are 2, 3 → find A³

A³ = P D³ P⁻¹

D³ =

Shortcut Tricks

• Distinct eigenvalues → directly diagonalizable
• Use diagonalization for Aⁿ problems
• Avoid multiplication → use Dⁿ
• Check eigenvector count carefully

Questions

Q1. When is matrix diagonalizable?

Answer: When it has n independent eigenvectors

Q2. Use of diagonalization?

Answer: Finding powers of matrix

Exam Tips

• Focus on eigenvalues first
• Check independence of eigenvectors
• Use diagonalization to save time
• Practice PYQs

🚀 Join Comprehensive Course

📢 GATE ESE PSU | ARJUNA Batch – Engineering Maths & General Aptitude

Prepare Engineering Mathematics & General Aptitude with Priyanka Sharma Ma’am (15+ Years Experience) at Studify+.

• ✔ Complete Engineering Mathematics coverage
• ✔ Complete General Aptitude preparation
• ✔ Live + Recorded Classes
• ✔ Concept clarity + practice questions
• ✔ 30 Years GATE PYQs (All Branches)
• ✔ Chapter & Subject Tests
• ✔ Designed for All GATE Branches

📚 Branches Covered:
ME | CE | EC | CS | EE | IN | MN | PE | XE | CH | PI | TF | AE | AG | GE & All

🔗 Explore Courses:
https://www.studifyplus.com/s/store

📱 Download App:
https://rb.gy/a3lwgu

📞 Contact:
WhatsApp / Call: 8200789441

✨ Build strong fundamentals and boost your GATE score!