Skew Symmetric Matrix - Properties, Theorems, and Examples

Skew Symmetric Matrix - Properties, Theorems, and Examples

What Is a Skew Symmetric Matrix?

A skew symmetric matrix (also called antisymmetric) is a square matrix A that satisfies

A^T = -A

where A^T is the transpose, the matrix formed by swapping rows and columns. Entry by entry, the condition reads a_{ij} = -a_{ji}: every entry is the negative of its mirror across the main diagonal.

A 3x3 example:

A = \begin{bmatrix} 0 & 2 & -5 \ -2 & 0 & 3 \ 5 & -3 & 0 \end{bmatrix}

Check one pair: a_{12} = 2 and a_{21} = -2, which are negatives, as required.

Variable glossary. a_{ij} is the entry in row i, column j. A^T is the transpose. The trace is the sum of the diagonal entries. A singular matrix has determinant zero (no inverse).

Why Are the Diagonal Entries Always Zero?

This is the first thing readers ask, and it falls straight out of the definition. For a diagonal entry, i = j, so the condition a_{ij} = -a_{ji} becomes:

a_{ii} = -a_{ii}

Add a_{ii} to both sides:

2a_{ii} = 0

a_{ii} = 0

Every diagonal entry must equal its own negative, and the only number that does is zero. So the diagonal is forced to be all zeros, and you never get to choose those entries. An immediate consequence: the trace (sum of the diagonal) is zero for every skew symmetric matrix.

What Are the Properties of a Skew Symmetric Matrix?

The defining condition A^T = -A propagates into a tidy set of properties.

The Decomposition Theorem: Every Square Matrix as Symmetric Plus Skew

Here is the result that gives skew symmetric matrices their importance. Any square matrix can be written as the sum of a symmetric matrix and a skew symmetric matrix, in exactly one way:

$$A = \underbrace{\frac{1}{2}(A + A^T)}{\text{symmetric}} + \underbrace{\frac{1}{2}(A - A^T)}{\text{skew symmetric}}$$

The first piece, \frac{1}{2}(A + A^T), is always symmetric. The second piece, \frac{1}{2}(A - A^T), is always skew symmetric. Add them and the transposes cancel back to A. This decomposition is why the skew symmetric part of a matrix appears throughout physics and engineering — it captures the "rotational" content of a transformation, while the symmetric part captures the "stretching" content.

Why Is the Determinant of an Odd-Order Skew Symmetric Matrix Zero?

A short, clean proof. Start from A^T = -A and take determinants of both sides:

∣A^T∣ = ∣-A∣

Use two facts: ∣A^T∣ = ∣A∣ (transposing does not change the determinant), and ∣-A∣ = (-1)^n∣A∣ for an n×n matrix (factoring -1 out of each of the n rows):

∣A∣ = (-1)^n∣A∣

When n is odd, (-1)^n = -1, so:

∣A∣ = -∣A∣

2∣A∣ = 0

∣A∣ = 0

So every odd-order skew symmetric matrix is singular. (For even n, (-1)^n = 1 and the equation gives ∣A∣ = ∣A∣, no information — even-order determinants need not be zero.)

Examples of a Skew Symmetric Matrix

Example 1

Is \begin{bmatrix} 0 & 6 \ -6 & 0 \end{bmatrix} skew symmetric?
Check a_{ij} = -a_{ji}. The diagonal is all zeros. Off-diagonal: a_{12} = 6, a_{21} = -6, which are negatives.
Final answer: Yes, it is skew symmetric.

Example 2

Is \begin{bmatrix} 0 & 4 \ 4 & 0 \end{bmatrix} skew symmetric?
This is the wrong-path-first example.

Wrong attempt. A student sees zeros on the diagonal and declares it skew symmetric, reasoning that the zero diagonal is the defining feature.

Why it is wrong. The zero diagonal is necessary but not sufficient. The off-diagonal entries must also be negatives of each other. Here a_{12} = 4 and a_{21} = 4 — equal, not opposite.
Final answer: No. It is symmetric, not skew symmetric — the zero diagonal alone is not enough.

Example 3

Find x so that \begin{bmatrix} 0 & x \ -7 & 0 \end{bmatrix} is skew symmetric.
The condition a_{12} = -a_{21} requires:

x = -(-7)

x = 7

Final answer: x = 7.

Example 4

Find the determinant of the 3×3 skew symmetric matrix \begin{bmatrix} 0 & 2 & -5 \ -2 & 0 & 3 \ 5 & -3 & 0 \end{bmatrix}.
The matrix is skew symmetric and has odd order (3), so the theorem says the determinant is zero. Final answer: ∣A∣ = 0.

Example 5

Decompose A = \begin{bmatrix} 2 & 5 \ 1 & 4 \end{bmatrix} into symmetric and skew symmetric parts.
Final answer: A = \begin{bmatrix} 2 & 3 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 2 \ -2 & 0 \end{bmatrix}.

Example 6

Verify that A^2 is symmetric for A = \begin{bmatrix} 0 & 3 \ -3 & 0 \end{bmatrix}.
Final answer: A^2 = \begin{bmatrix} -9 & 0 \ 0 & -9 \end{bmatrix}, which is symmetric, as the property predicts.

Why Skew Symmetric Matrices Matter: "The math of pure rotation"

Skew symmetric matrices exist because rotation and the cross product needed an algebraic home. When a transformation rotates without stretching, a rigid spin, its generator is skew symmetric, because rotation has no "diagonal" stretching content, only the off-diagonal mixing of axes. The forced-zero diagonal is the algebra noticing that pure rotation cannot scale.

Skew Symmetric Matrix - Where Intuition Breaks

Mistake 1: Believing the zero diagonal alone makes a matrix skew symmetric

Mistake 2: Mixing up the sign in the decomposition theorem

Mistake 3: Assuming every skew symmetric determinant is zero

The Mathematicians Behind the Skew Symmetric Matrix

Arthur Cayley developed the algebra of matrices, including the symmetric and skew symmetric classifications. Ferdinand Georg Frobenius extended the theory of bilinear and quadratic forms that underpins the symmetric-plus-skew decomposition.

Skew Symmetric Matrix - In Five Bullets