Editing Skew-symmetric matrix (section)

=== Vector space structure ===
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 <math display=inline>n \times n</math> skew-symmetric matrices has [[Dimension of a vector space|dimension]] <math display=inline>\frac{1}{2}n(n - 1).</math>

Let <math>\mbox{Mat}_n</math> denote the space of <math display=inline>n \times n</math> matrices. A skew-symmetric matrix is determined by <math display=inline>\frac{1}{2}n(n - 1)</math> scalars (the number of entries above the [[main diagonal]]); a [[symmetric matrix]] is determined by  <math display=inline>\frac{1}{2}n(n + 1)</math> scalars (the number of entries on or above the main diagonal). Let <math display=inline>\mbox{Skew}_n</math> denote the space of <math display=inline>n \times n</math> skew-symmetric matrices and <math display=inline>\mbox{Sym}_n</math> denote the space of <math display=inline>n \times n</math> symmetric matrices. If <math display=inline>A \in \mbox{Mat}_n</math> then
<math display="block">A = \tfrac{1}{2}\left(A - A^\mathsf{T}\right) + \tfrac{1}{2}\left(A + A^\mathsf{T}\right).</math>

Notice that <math display=inline>\frac{1}{2}\left(A - A^\textsf{T}\right) \in \mbox{Skew}_n</math> and <math display=inline>\frac{1}{2}\left(A + A^\textsf{T}\right) \in \mbox{Sym}_n.</math> This is true for every [[square matrix]] <math display=inline>A</math> with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2. Then, since <math display=inline>\mbox{Mat}_n = \mbox{Skew}_n + \mbox{Sym}_n</math> and <math display=inline>\mbox{Skew}_n \cap \mbox{Sym}_n = \{0\},</math> 
<math display=block>\mbox{Mat}_n = \mbox{Skew}_n \oplus \mbox{Sym}_n,</math>
where <math>\oplus</math> denotes the [[Direct sum of modules|direct sum]].

Denote by <math display=inline>\langle \cdot, \cdot \rangle</math> the standard [[inner product]] on <math>\R^n.</math> The real <math>n \times n</math> matrix <math display=inline>A</math> is skew-symmetric if and only if
<math display=block>\langle Ax,y \rangle = - \langle x, Ay\rangle \quad \text{ for all } x, y \in \R^n.</math>

This is also equivalent to <math display=inline>\langle x, Ax \rangle = 0</math> for all <math>x \in \R^n</math> (one implication being obvious, the other a plain consequence of <math display=inline>\langle x + y, A(x + y)\rangle = 0</math> for all <math>x</math> and <math>y</math>).

Since this definition is independent of the choice of [[Basis (linear algebra)|basis]], skew-symmetry is a property that depends only on the [[linear operator]] <math>A</math> and a choice of [[inner product]].

<math>3 \times 3</math> skew symmetric matrices can be used to represent [[cross product]]s as matrix multiplications.

Furthermore, if <math>A</math> is a skew-symmetric (or [[Skew-Hermitian matrix|skew-Hermitian]]) matrix, then <math>x^T A x = 0</math> for all <math>x \in \C^n</math>.