Skew symmetry

In mathematics, and in particular linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose is its negation; that is, it satisfies the condition −A = A^T. If the entry in the i th row and j th column is a_ij, i.e. A = (a_ij) then the skew symmetric condition is a_ij = −a_ji. For example, the following matrix is skew-symmetric:

We assume that the underlying field is not of characteristic 2: that is, that 1 + 1 ≠ 0 where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Otherwise, a skew-symmetric matrix is just the same thing as a symmetric matrix.

Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. Its dimension is n(n−1)/2.

Let Mat_n denote the space of n × n matrices. A skew-symmetric matrix is determined by n(n − 1)/2 scalars (the number of entries above the main diagonal); a symmetric matrix is determined by n(n + 1)/2 scalars (the number of entries on or above the main diagonal). Let Skew_n denote the space of n × n skew-symmetric matrices and Sym_n denote the space of n × n symmetric matrices. If A ∈ Mat_n then

Notice that ½(A − A^T) ∈ Skew_n and ½(A + A^T) ∈ Sym_n. This is true for every square matrix A with entries from any field whose characteristic is different from 2. Then, since Mat_n = Skew_n + Sym_n and Skew_n ∩ Sym_n = {0},

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