Skew symmetric matrix

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; i.e., A = (a_ij), then the skew-symmetric condition is a_ji = −a_ij. For example, the following matrix is skew-symmetric:

Throughout, we assume that all matrix entries belong to a field ${\displaystyle \mathbb {F} }$ whose characteristic is not equal to 2: that is, we assume that 1 + 1 ≠ 0, where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix.

As a result of the first two properties above, the set of all skew-symmetric matrices of a fixed size forms a vector space. The space of ${\displaystyle n\times n}$ skew-symmetric matrices has dimension n(n−1)/2.

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