Skew-symmetric matrix

In linear algebra, a square matrix A is said to be skew-symmetric or antisymmetric if its transpose is also its negative; that is, it satisfies the equation:

A^T = −A

or in component form, if A = (a_i,j):

a_i,j = − a_j,i for all i and j

For example, the following matrix is skew-symmetric:

{\begin{bmatrix}0&2\\-2&0\end{bmatrix}}

All main diagonal entries of a skew-symmetric matrix have to be zero, and so the trace is zero.

Infinitesimal rotations

The skew-symmetric n×n matrices form a vector space of dimension (n² - n)/2. This is the tangent space to the orthogonal group O(n). In a sense, then, skew-symmetric matrices can be thought of as "infinitesimal rotations".

In fact, the skew-symmetric n-by-n matrices form a Lie algebra using the commutator Lie bracket

[A,B]=AB-BA\,

and this is the Lie algebra associated to the Lie group O(n).

A matrix G is orthogonal and has determinant 1, i.e., it is a member of that connected component of the orthogonal group in which the identity element lies, precisely if for some skew-symmetric matrix A we have

G=\exp(A)=\sum _{n=0}^{\infty }{\frac {A^{n}}{n!}}.

Alternating forms

An alternating form φ on a vector space V over a field K is defined (if K doesn't have characteristic 2) to be a bilinear form

φ : V x V -> K

such that

φ(v,w) = -φ(w,v).

Such a φ will be represented by a skew-symmetric matrix, once a basis of V is chosen; and conversely an nxn skew-symmetric matrix A on Kⁿ gives rise to an alternating form x^τ.A.x.