Redirected from Skew-symmetric
or in component form, if A = (ai,j):
For example, the following matrix is skew-symmetric:
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
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
See also symmetric matrix.
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