A square matrix 
is antihermitian if it satisfies
 | (1) |
where 
is the adjoint. For example, the matrix
![[i 1+i 2i; -1+i 5i 3; 2i -3 0]](http://mathworld.wolfram.com/images/equations/AntihermitianMatrix/NumberedEquation2.svg) | (2) |
is an antihermitian matrix. Antihermitian matrices are often called "skew Hermitian matrices" by mathematicians.
A matrix 
can be tested to see if it is antihermitian in the Wolfram Language using AntihermitianMatrixQ[m].
The set of 
antihermitian matrices is a vector space, and the commutator
![[A,B]=AB-BA](http://mathworld.wolfram.com/images/equations/AntihermitianMatrix/NumberedEquation3.svg) | (3) |
of two antihermitian matrices is antihermitian. Hence, the antihermitian matrices are a Lie algebra, which is related to the Lie group of unitary matrices. In particular, suppose 
is a path of unitary matrices through 
, i.e.,
 | (4) |
for all 
, where 
is the adjoint and 
is the identity matrix. The derivative at 
of both sides must be equal so
 | (5) |
That is, the derivative of 
at the identity must be antihermitian.
The matrix exponential map of an antihermitian matrix is a unitary matrix.
