An antisymmetric matrix, also known as a skew-symmetric or antimetric matrix, is a square matrix that satisfies the identity
 | (1) |
where 
is the matrix transpose. For example,
![A=[0 -1; 1 0]](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation2.svg) | (2) |
is antisymmetric.
A matrix 
may be tested to see if it is antisymmetric in the Wolfram Language using AntisymmetricMatrixQ[m].
In component notation, this becomes
 | (3) |
Letting 
, the requirement becomes
 | (4) |
so an antisymmetric matrix must have zeros on its diagonal. The general 
antisymmetric matrix is of the form
![[0 a_(12) a_(13); -a_(12) 0 a_(23); -a_(13) -a_(23) 0].](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation5.svg) | (5) |
Applying 
to both sides of the antisymmetry condition gives
 | (6) |
Any square matrix can be expressed as the sum of symmetric and antisymmetric parts. Write
 | (7) |
But
![A=[a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | | ... |; a_(n1) a_(n2) ... a_(nn)]](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation8.svg) | (8) |
![A^(T)=[a_(11) a_(21) ... a_(n1); a_(12) a_(22) ... a_(n2); | | ... |; a_(1n) a_(2n) ... a_(nn)],](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation9.svg) | (9) |
so
![A+A^(T)=[2a_(11) a_(12)+a_(21) ... a_(1n)+a_(n1); a_(12)+a_(21) 2a_(22) ... a_(2n)+a_(n2); | | ... |; a_(1n)+a_(n1) a_(2n)+a_(n2) ... 2a_(nn)],](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation10.svg) | (10) |
which is symmetric, and
![A-A^(T)=[0 a_(12)-a_(21) ... a_(1n)-a_(n1); -(a_(12)-a_(21)) 0 ... a_(2n)-a_(n2); | | ... |; -(a_(1n)-a_(n1)) -(a_(2n)-a_(n2)) ... 0],](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation11.svg) | (11) |
which is antisymmetric.
All 
antisymmetric matrices of odd dimension are singular. This follows from the fact that
 | (12) |
So, by the properties of determinants,
 |  |  | (13) |
 |  |  | (14) |
Therefore, if 
is odd, then
 | (15) |
thus proving all antisymmetric matrices of odd dimension are singular.
The set of 
antisymmetric matrices is denoted 
. 
is a vector space, and the commutator
![[A,B]=AB-BA](https://mathworld.wolfram.com/images/equations/AntisymmetricMatrix/NumberedEquation14.svg) | (16) |
of two antisymmetric matrices is antisymmetric. Hence, the antisymmetric matrices are a Lie algebra, which is related to the Lie group of orthogonal matrices. In particular, suppose 
is a path of orthogonal matrices through 
, i.e., 
for all 
. The derivative at 
of both sides must be equal so 
. That is, the derivative of 
at the identity must be an antisymmetric matrix.
The matrix exponential map of an antisymmetric matrix is an orthogonal matrix.
