## Antisymmetric Matrix

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.