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Concept# Skew-symmetric matrix

Summary

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_{ij} denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to
Example
The matrix
:A =
\begin{bmatrix}
0 & 2 & -45 \
-2 & 0 & -4 \
45 & 4 & 0
\end{bmatrix}
is skew-symmetric because
: -A =
\begin{bmatrix}
0 & -2 & 45 \
2 & 0 & 4 \
-45 & -4 & 0
\end{bmatrix} = A^\textsf{T}
.
Properties
Throughout, we assume that all matrix entries belong to a field \mathbb{F} whose characteristic is not equal to 2. That is, we assume that 1 + 1 ≠ 0, where 1 denotes the multiplicat

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