Hermitian, Skew-Hermitian & Unitary Matrices – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

Hermitian, Skew-Hermitian, and Unitary matrices are important concepts in Linear Algebra under Engineering Mathematics. These topics are frequently asked in GATE, ESE, AE/JE exams, especially in complex matrix problems.

Basic Concepts

For complex matrices, we use conjugate transpose (A^H), which means:

A^H = (A^T)̅ → Transpose + Complex Conjugate

Hermitian Matrix

A matrix A is Hermitian if:

A^H = A

Example

A =

A^H = A → Hermitian Matrix

Important Property

• Diagonal elements are always real

Skew-Hermitian Matrix

A matrix A is Skew-Hermitian if:

A^H = −A

Example

A =

A^H = −A → Skew-Hermitian Matrix

Important Property

• Diagonal elements are purely imaginary or zero

Unitary Matrix

A matrix A is Unitary if:

A^HA = I

A⁻¹ = A^H

Example

A =

A^HA = I → Unitary Matrix

Important Properties

• Hermitian ⇒ Eigenvalues are real
• Skew-Hermitian ⇒ Eigenvalues are purely imaginary
• Unitary ⇒ |Eigenvalues| = 1
• Unitary ⇒ Inverse = Conjugate Transpose

Solved Examples

Example 1

Check if matrix is Hermitian:

Answer: Hermitian

Example 2

Check if matrix is Skew-Hermitian:

Answer: Skew-Hermitian

Example 3

If A is unitary, find A⁻¹

Answer: A^H

Example 4

If A is Hermitian, nature of eigenvalues?

Answer: Real

Shortcut Tricks

• Hermitian → A = A^H
• Skew-Hermitian → A = −A^H
• Unitary → A^HA = I
• Diagonal real → Hermitian
• Diagonal imaginary → Skew-Hermitian

Questions

Q1. Eigenvalues of Hermitian matrix?

Answer: Real

Q2. If A is unitary, determinant?

Answer: |det(A)| = 1

Exam Tips

• Focus on conjugate transpose concept
• Memorize key properties
• Avoid lengthy calculations
• Practice eigenvalue-based questions

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