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

# Hermitian and skew-Hermitian matrices

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2068 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define Hermitian and skew-Hermitian matrices. Show that every square matrix can be uniquely expressed as the sum of a Hermitian and a skew-Hermitian matrix.

## Short answer

A matrix is Hermitian if A* = A (conjugate transpose) and skew-Hermitian if A* = −A. Every square matrix A can be uniquely expressed as A = P + Q with P = ½(A+A*) Hermitian and Q = ½(A−A*) skew-Hermitian.

## Answer

#### Definitions
Let A* = (Ā)ᵀ be the conjugate transpose of a complex square matrix A.

- A is **Hermitian** if A* = A (aᵢⱼ = conj(aⱼᵢ); the diagonal entries are real). Example: [3  2−i; 2+i  5].
- A is **skew-Hermitian** if A* = −A (aᵢⱼ = −conj(aⱼᵢ); the diagonal entries are zero or purely imaginary). Example: [2i  1+i; −1+i  0].

#### Existence
Define P = ½(A + A*) and Q = ½(A − A*). Using (A*)* = A and (kA)* = k̄A* with k = ½ real:

- P* = ½(A* + A) = P, so P is Hermitian.
- Q* = ½(A* − A) = −Q, so Q is skew-Hermitian.
- P + Q = A.

#### Uniqueness
Suppose A = R + S with R Hermitian and S skew-Hermitian. Taking the conjugate transpose:

```text
A* = R* + S* = R − S
```

Adding and subtracting with A = R + S:

```text
A + A* = 2R  ⇒  R = ½(A + A*) = P
A − A* = 2S  ⇒  S = ½(A − A*) = Q
```

So the decomposition is unique.

**Example.** A = [1  2+i; 3  i]. Then A* = [1  3; 2−i  −i].
P = ½(A + A*) = [1  (5+i)/2; (5−i)/2  0] (Hermitian), and
Q = ½(A − A*) = [0  (−1+i)/2; (1+i)/2  i] (skew-Hermitian), with P + Q = A. ∎

## Check yourself

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

1. Which condition defines a skew-Hermitian matrix A?
   A. A* = A
   B. A* = -A
   C. Aᵀ = A
   D. A* = A^(-1)
   Answer: B. Skew-Hermitian means the conjugate transpose equals the negative of the matrix, A* = -A. (Key point: Definitions)

2. What can be said about the diagonal entries of a Hermitian matrix?
   A. They are real
   B. They are purely imaginary
   C. They are all zero
   D. They are equal to each other
   Answer: A. aᵢᵢ = conj(aᵢᵢ) forces each diagonal entry to be real. (Key point: Diagonal entries)

3. Which choice of P and Q gives A = P + Q with P Hermitian and Q skew-Hermitian?
   A. P = ½(A + Aᵀ), Q = ½(A − Aᵀ)
   B. P = ½(A − A*), Q = ½(A + A*)
   C. P = A + A*, Q = A − A*
   D. P = ½(A + A*), Q = ½(A − A*)
   Answer: D. P* = P and Q* = -Q for these choices, and P + Q = A. (Key point: Existence)

4. In the uniqueness proof, if A = R + S (R Hermitian, S skew-Hermitian), what is A*?
   A. R + S
   B. −R + S
   C. R − S
   D. −R − S
   Answer: C. A* = R* + S* = R − S, so A + A* = 2R and A − A* = 2S, forcing R = P and S = Q. (Key point: Uniqueness)

## Close variants set in other exams

- [Hermitian plus skew-Hermitian decomposition](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahermitian-plus-skew-hermitian-decomposition) — Mathematics III, chapter 1, asked 5×

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