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

# Symmetric plus skew-symmetric decomposition

Mathematics III · Chapter 1: Determinants and Matrices

- Asked in: 2080 Bhadra, 1+4 marks
- Video: [Any square matrix can be written as sum of Symmetric and Skew Symmetric matrix and that too UNIQUELY](https://www.youtube.com/watch?v=HSl628sH1FE) — Dr. Mathaholic
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/symmetric-plus-skew-symmetric-decomposition-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1

## Question

Define the transpose of a matrix. Prove that every square matrix can be expressed as a sum of a symmetric matrix and a skew-symmetric matrix uniquely.

## Short answer

The transpose Aᵀ is obtained by interchanging rows and columns. Any square matrix A can be uniquely expressed as P + Q, where P = ½(A + Aᵀ) is symmetric and Q = ½(A − Aᵀ) is skew-symmetric. Uniqueness follows by solving A = R + S with Rᵀ = R and Sᵀ = −S.

## Answer

The **transpose** of a matrix A = [aᵢⱼ] of order m × n is the matrix Aᵀ of order n × m obtained by interchanging its rows and columns, so (Aᵀ)ᵢⱼ = aⱼᵢ. A matrix is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A.

A square matrix A can always be written as A = P + Q, where P = ½(A + Aᵀ) is symmetric and Q = ½(A − Aᵀ) is skew-symmetric, and this representation is unique.

#### Existence
Let A be any square matrix. Define

```text
P = ½(A + Aᵀ),    Q = ½(A − Aᵀ)
```

- Pᵀ = ½(Aᵀ + (Aᵀ)ᵀ) = ½(Aᵀ + A) = P, so P is symmetric.
- Qᵀ = ½(Aᵀ − (Aᵀ)ᵀ) = ½(Aᵀ − A) = −Q, so Q is skew-symmetric.
- P + Q = ½(A + Aᵀ) + ½(A − Aᵀ) = A.

Hence A is the sum of a symmetric and a skew-symmetric matrix.

#### Uniqueness
Suppose A = R + S is another such decomposition, where R is symmetric (Rᵀ = R) and S is skew-symmetric (Sᵀ = −S). Take the transpose of A:

```text
Aᵀ = Rᵀ + Sᵀ = R − S
```

Now add and subtract A = R + S and Aᵀ = R − S:

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

So R = P and S = Q. The decomposition is unique.

**Example.** For A = [1 2; 4 3]: Aᵀ = [1 4; 2 3], P = [1 3; 3 3], Q = [0 −1; 1 0], and P + Q = A.

Therefore every square matrix is uniquely the sum of a symmetric and a skew-symmetric matrix. ∎

## Check yourself

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

1. Which choice of P and Q gives A = P + Q with P symmetric and Q skew-symmetric?
   A. P = (A + A^T)/2, Q = (A - A^T)/2
   B. P = A + A^T, Q = A - A^T
   C. P = (A + A^T)/2, Q = (A^T - A)/2 with A^T on the right only
   D. P = (A - A^T)/2, Q = (A + A^T)/2
   Answer: A. P = (A+A^T)/2 is unchanged by transposing, Q = (A-A^T)/2 changes sign, and P + Q = A. (Key point: Choice of P and Q)

2. Why is Q = (A - A^T)/2 skew-symmetric?
   A. Q^T = (A - A^T)/2 = Q
   B. Q^T = (A^T + A)/2 = P
   C. Q^T = (A^T - A)/2 = -Q
   D. Q^T = 0
   Answer: C. Transposing swaps A and A^T, so the sign of the bracket flips, giving Q^T = -Q. (Key point: Showing Q is skew-symmetric)

3. In the uniqueness proof, if A = R + S (R symmetric, S skew-symmetric), what is A^T?
   A. R + S
   B. R - S
   C. -R + S
   D. -R - S
   Answer: B. R^T = R and S^T = -S, so A^T = R - S. Adding and subtracting with A = R + S gives R and S. (Key point: Uniqueness argument)

4. What does the transpose of an m x n matrix look like?
   A. An m x n matrix with each entry negated
   B. An m x n matrix with only the diagonal kept
   C. An n x m matrix with every entry inverted
   D. An n x m matrix with rows and columns interchanged
   Answer: D. A^T has (i,j) entry a_ji, so it is n x m. (Key point: Definition of transpose)

## Close variants set in other exams

- [Symmetric plus skew-symmetric decomposition](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asymmetric-plus-skew-symmetric-decomposition) — Mathematics III, chapter 1, asked 8×

## Related in Mathematics III

- [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×
- [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×