Antihermitian Matrix

An antihermitian matrix is a square matrix that satisfies

(1)

where is the conjugate transpose. For example, the matrix

(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 real vector space, and the commutator

(3)

of two antihermitian matrices is antihermitian. Hence, the antihermitian matrices form a Lie algebra over the real numbers, which is related to the Lie group of unitary matrices. In particular, suppose is a differentiable path of unitary matrices through , i.e.,

(4)

for all , where is the conjugate transpose and is the identity matrix. The derivative at of both sides must be equal so

(5)

That is, the derivative of at the identity matrix must be antihermitian.

The matrix exponential of an antihermitian matrix is a unitary matrix.