Encyclopedia > Skew-symmetric

  Article Content

Skew-symmetric matrix

Redirected from Skew-symmetric

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:

AT = -A

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

ai,j = - aj,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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Battle Creek, Michigan

... is 3.04. In the city the population is spread out with 27.2% under the age of 18, 8.7% from 18 to 24, 29.5% from 25 to 44, 21.0% from 45 to 64, and 13.5% who are 65 ...

 
 
 
This page was created in 25.9 ms