An antisymmetric matrix is a Matrix which satisfies the identity
|
(1) |
where
is the Matrix Transpose. In component notation, this becomes
|
(2) |
Letting , the requirement becomes
|
(3) |
so an antisymmetric matrix must have zeros on its diagonal. The general antisymmetric matrix is of the form
|
(4) |
Applying
to both sides of the antisymmetry
condition gives
|
(5) |
Any Square Matrix can be expressed as the sum of symmetric and antisymmetric parts. Write
|
(6) |
But
|
(7) |
|
(8) |
so
|
(9) |
which is symmetric, and
|
(10) |
which is antisymmetric.
See also Skew Symmetric Matrix, Symmetric Matrix
© 1996-9 Eric W. Weisstein
1999-05-25