Skew-symmetric Matrix - Properties

Properties

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). If Skew_n denotes the space of n × n skew-symmetric matrices and Sym_n denotes the space of n × n symmetric matrices and then since Mat_n = Skew_n + Sym_n and Skew_n ∩ Sym_n = {0}, i.e.

where ⊕ denotes the direct sum. Let A ∈ Mat_n then

Notice that ½(A − AT) ∈ Skew_n and ½(A + AT) ∈ Sym_n. This is true for every square matrix A with entries from any field whose characteristic is different from 2.

As to equivalent conditions, notice that the relation of skew-symmetricity, A=-AT, holds for a matrix A if and only if one has xTAy =-yTAx for all vectors x and y. This is also equivalent to xTAx=0 for all x (one implication being obvious, the other a plain consequence of (x+y)TA(x+y)=0 for all x and y).

All main diagonal entries of a skew-symmetric matrix must be zero, so the trace is zero. If A = (a_ij) is skew-symmetric, a_ij = −a_ji; hence a_ii = 0.

3x3 skew symmetric matrices can be used to represent cross products as matrix multiplications.