Skew-symmetric matrixIn linear algebra, a square matrix A is said to be skew-symmetric if its transpose is also its negative; that is, it satisfies the equation:
- AT = -A
- ai,j = - aj,i for all i and j
The skew-symmetric n-by-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 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