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
© 19969 Eric W. Weisstein
19990525