---
title: "Hermitian and skew-Hermitian matrices"
subject: "Mathematics III"
chapter: 1
key: "semester_3/mathematics_iii:hermitian-and-skew-hermitian-matrices"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/hermitian-and-skew-hermitian-matrices
site: BCT Engineering
---

# Hermitian and skew-Hermitian matrices

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2078 Bhadra, 5 marks
- Video: [Hermitian Matrix & Skew-Hermitian Matrix | Concept & Questions based on MSQs - MSQs Series By GP Sir](https://www.youtube.com/watch?v=ijGjvktg7vA) — Dr.Gajendra Purohit - GATE / IIT JAM / CSIR NET
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/hermitian-and-skew-hermitian-matrices
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define Hermitian and skew-Hermitian matrices of a square complex matrix. If A is any square matrix, prove that A + A* is Hermitian and A − A* is skew-Hermitian.

## Short answer

Hermitian: A* = A; skew-Hermitian: A* = −A. Then (A+A*)* = A*+A = A+A*, so Hermitian; (A−A*)* = A*−A = −(A−A*), so skew-Hermitian.

## Answer

#### Definitions
Let A* (also written Aᶿ) denote the **conjugate transpose** of A: A* = (Ā)ᵀ, i.e. take the complex conjugate of every entry and transpose.

- A square matrix A is **Hermitian** if A* = A, i.e. aᵢⱼ = conj(aⱼᵢ). Its diagonal entries are real. Example: [2  1−i; 1+i  3].
- A square matrix A is **skew-Hermitian** if A* = −A, i.e. aᵢⱼ = −conj(aⱼᵢ). Its diagonal entries are purely imaginary or zero. Example: [i  2+i; −2+i  0].

#### Proof that A + A* is Hermitian and A − A* is skew-Hermitian
We use the properties (A + B)* = A* + B*, (A − B)* = A* − B*, and (A*)* = A.

**A + A* is Hermitian:**
```text
(A + A*)* = A* + (A*)* = A* + A = A + A*
```
So (A + A*)* = A + A*, i.e. A + A* is Hermitian.

**A − A* is skew-Hermitian:**
```text
(A − A*)* = A* − (A*)* = A* − A = −(A − A*)
```
So (A − A*)* = −(A − A*), i.e. A − A* is skew-Hermitian. ∎

**Consequence:** A = ½(A + A*) + ½(A − A*), which expresses A as the sum of a Hermitian and a skew-Hermitian matrix.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is A*, the conjugate transpose of A?
   A. The transpose of A only
   B. The conjugate of A only
   C. The inverse of A
   D. Conjugate every entry, then transpose
   Answer: D. A* = (conjugate of A) transposed. (Key point: Conjugate transpose)

2. Which pair of conditions defines Hermitian and skew-Hermitian?
   A. A* = A and A* = -A
   B. A^T = A and A^T = -A
   C. A* = A^-1 and A* = -A^-1
   D. A* = -A and A* = A
   Answer: A. Hermitian means A* = A; skew-Hermitian means A* = -A. (Key point: Definitions)

3. Which rule is used to show (A + A*)* = A + A*?
   A. (AB)* = A*B*
   B. (A + B)* = A* + B* and (A*)* = A
   C. (A*)* = -A
   D. |A*| = |A|
   Answer: B. (A + A*)* = A* + (A*)* = A* + A, which equals A + A*. (Key point: Proof for A + A*)

4. What is (A - A*)* equal to?
   A. A - A*
   B. A + A*
   C. A* - A = -(A - A*)
   D. 0
   Answer: C. (A - A*)* = A* - A = -(A - A*), so it is skew-Hermitian. (Key point: Proof for A - A*)

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