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Antisymmetric Matrix — Definition, Formula & Examples

An antisymmetric matrix (also called a skew-symmetric matrix) is a square matrix that equals the negative of its own transpose. This forces every diagonal entry to be zero.

A square matrix AA is antisymmetric if A⊤=−AA^\top = -A, which means each entry satisfies aij=−ajia_{ij} = -a_{ji} for all indices ii and jj. In particular, setting i=ji = j gives aii=−aiia_{ii} = -a_{ii}, so every diagonal entry must equal zero.

Key Formula

A⊤=−A⟺aij=−aji for all i, jA^\top = -A \quad \Longleftrightarrow \quad a_{ij} = -a_{ji} \text{ for all } i,\, j
Where:
  • AA = A square matrix
  • A⊤A^\top = The transpose of A
  • aija_{ij} = The entry in row i, column j of A

How It Works

To check whether a matrix is antisymmetric, compute its transpose and see if it equals the negative of the original. Equivalently, verify that aij=−ajia_{ij} = -a_{ji} for every pair of indices. Any real antisymmetric matrix has purely imaginary (or zero) eigenvalues, and its determinant is always non-negative when the matrix is of even order. For odd-order real antisymmetric matrices, the determinant is exactly zero, meaning they are always singular.

Worked Example

Problem: Determine whether the matrix A is antisymmetric, where A = [[0, 3, -1], [-3, 0, 7], [1, -7, 0]].
Step 1: Compute the transpose of A by reflecting entries across the main diagonal.
A⊤=[0−3130−7−170]A^\top = \begin{bmatrix} 0 & -3 & 1 \\ 3 & 0 & -7 \\ -1 & 7 & 0 \end{bmatrix}
Step 2: Compute the negative of A.
−A=[0−3130−7−170]-A = \begin{bmatrix} 0 & -3 & 1 \\ 3 & 0 & -7 \\ -1 & 7 & 0 \end{bmatrix}
Step 3: Compare the two results. Since every entry of A^T matches the corresponding entry of -A, the condition is satisfied.
A⊤=−A✓A^\top = -A \quad \checkmark
Answer: Yes, A is antisymmetric. Note that all diagonal entries are zero and each off-diagonal pair satisfies aij=−ajia_{ij} = -a_{ji}.

Why It Matters

Antisymmetric matrices appear in physics when representing cross products and angular velocity. In differential geometry and mechanics, the Lie algebra of the rotation group SO(n)SO(n) consists entirely of antisymmetric matrices, making them essential for describing rotational motion.

Common Mistakes

Mistake: Forgetting that the diagonal entries must be zero.
Correction: The condition aii=−aiia_{ii} = -a_{ii} forces aii=0a_{ii} = 0 for every diagonal entry. If any diagonal entry is nonzero, the matrix cannot be antisymmetric.