# Definition of skew-symmetric matrix

Put content here.**Definition.** A matrix $A$ is *skew-symmetric* (or *antisymmetric*) if $A^T = -A$.
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Equivalently, $a_{ij} = -a_{ji}$ for all $i, j$. This implies:
- All diagonal entries are zero: $a_{ii} = 0$
- $A$ is always square
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**Example.** $A = \begin{pmatrix} 0 & 3 & -1 \\ -3 & 0 & 2 \\ 1 & -2 & 0 \end{pmatrix}$ is skew-symmetric.
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**Properties:**
- Every square matrix can be uniquely written as the sum of a symmetric and a skew-symmetric matrix: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$
- The eigenvalues of a real skew-symmetric matrix are purely imaginary (or zero)
- For any vector $x$: $x^T A x = 0$

# Parents

* Symmetric matrices