A square matrix is antihermitian if it satisfies
(1)
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where is the adjoint. For example, the matrix
(2)
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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
(3)
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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)
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for all , where is the adjoint and is the identity matrix. The derivative at of both sides must be equal so
(5)
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That is, the derivative of at the identity must be antihermitian.
The matrix exponential map of an antihermitian matrix is a unitary matrix.