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

# Hermitian and skew-Hermitian matrices

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2069 Chaitra, 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Show that the matrix Bᶿ A B is Hermitian or skew-Hermitian according as A is Hermitian or skew-Hermitian (Bᶿ denotes the conjugate transpose of B).

## Short answer

Conjugate transpose gives (BᶿAB)ᶿ = BᶿAᶿB. If Aᶿ = A, result equals BᶿAB (Hermitian). If Aᶿ = −A, result equals −BᶿAB (skew-Hermitian).

## Answer

Let Bᶿ denote the conjugate transpose of B. We use the rules (XY)ᶿ = YᶿXᶿ and (Xᶿ)ᶿ = X. A matrix H is Hermitian if Hᶿ = H and skew-Hermitian if Hᶿ = −H.

#### Take the conjugate transpose of BᶿAB
```text
(BᶿAB)ᶿ = Bᶿ Aᶿ (Bᶿ)ᶿ = Bᶿ Aᶿ B
```
(the order is reversed, and (Bᶿ)ᶿ = B).

#### Case 1: A is Hermitian (Aᶿ = A)
```text
(BᶿAB)ᶿ = Bᶿ A B
```
So BᶿAB equals its own conjugate transpose, i.e. BᶿAB is **Hermitian**.

#### Case 2: A is skew-Hermitian (Aᶿ = −A)
```text
(BᶿAB)ᶿ = Bᶿ(−A)B = −(BᶿAB)
```
So BᶿAB is **skew-Hermitian**. ∎

Hence BᶿAB is Hermitian or skew-Hermitian according as A is Hermitian or skew-Hermitian. (B can be any matrix for which the product is defined; no condition on B is needed.)

## Check yourself

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

1. (B*AB)* equals:
   A. B*A*B
   B. BA*B*
   C. B*A*B*
   D. A*
   Answer: A. Reverse order and use (B*)* = B: (B*AB)* = B*A*B. (Key point: Conjugate transpose of product)

2. If A* = A, then (B*AB)* = B*AB, so B*AB is:
   A. Skew-Hermitian
   B. Hermitian
   C. Orthogonal
   D. Zero
   Answer: B. It equals its own conjugate transpose. (Key point: Hermitian case)

3. If A* = -A, then (B*AB)* is:
   A. B*AB
   B. B*A*B*
   C. -(B*AB), so skew-Hermitian
   D. Undefined
   Answer: C. B*(-A)B = -(B*AB). (Key point: Skew case)

4. What must B satisfy?
   A. B Hermitian
   B. B invertible
   C. B unitary
   D. Nothing beyond the product being defined
   Answer: D. The result holds for any B for which the product is defined. (Key point: Condition on B)

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