Skew-symmetric matrix

In 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:

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:

<math>\begin{bmatrix}

0 & 2 \\ -2 & 0 \end{bmatrix}</math>

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

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

<math>[A,B] = AB - BA\,</math>

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

<math>G=\exp(A)=\sum_{n=0}^\infty \frac{A^n}{n!}.</math>

See also symmetric matrix.