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

# Hermitian plus skew-Hermitian decomposition

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Baishakh, 5 marks; 2076 Chaitra, 5 marks; 2071 Shrawan, 5 marks; 2067 Asar (asked 4 times)
- 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-plus-skew-hermitian-decomposition
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

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

## Short answer

Write A = P + Q with P = ½(A + A*) Hermitian and Q = ½(A − A*) skew-Hermitian. The decomposition is unique because any representation implies P = ½(A + A*) and Q = ½(A − A*).

## Answer

A complex square matrix A can be written as A = P + Q, where P = ½(A + A*) is Hermitian and Q = ½(A − A*) is skew-Hermitian, and the decomposition is unique. Here A* = (Ā)ᵀ is the conjugate transpose.

A matrix H is Hermitian if H* = H, and skew-Hermitian if H* = −H.

#### 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. What is A* for a complex matrix A?
   A. The conjugate of A only
   B. The transpose of A only
   C. The transpose of the conjugate of A
   D. The negative of A^T
   Answer: C. A* = (conj A)^T. (Key point: Conjugate transpose)

2. Which P and Q give a Hermitian plus skew-Hermitian split of A?
   A. P = (A + A*)/2, Q = (A - A*)/2
   B. P = (A + A^T)/2, Q = (A - A^T)/2
   C. P = (A + A*)/2, Q = (A* - A)/2 with Q Hermitian
   D. P = A + A*, Q = A - A*
   Answer: A. P* = P and Q* = -Q, and P + Q = A. (Key point: Choice of P and Q)

3. Why can the factor 1/2 be pulled out when taking the conjugate transpose?
   A. Because (kA)* = -kA*
   B. It is real, so (kA)* = conj(k) A* = kA*
   C. Because 1/2 is the conjugate of 2
   D. Because conjugate transpose is not affected by scalars at all, even complex ones
   Answer: B. A real scalar equals its own conjugate. (Key point: Why 1/2 is harmless)

4. If A = R + S with R Hermitian and S skew-Hermitian, what is A*?
   A. R + S
   B. -R + S
   C. -R - S
   D. R - S
   Answer: D. R* = R and S* = -S; adding and subtracting with A gives R = P and S = Q. (Key point: Uniqueness)

## Close variants set in other exams

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

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