A square matrix is antihermitian if it satisfies

(1)

where
is the adjoint . 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 vector space , and the
commutator

(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 .

See also Adjoint ,

Antisymmetric Part ,

Hermitian Matrix ,

Unitary
Matrix
Portions of this entry contributed by Todd
Rowland

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Cite this as:
Rowland, Todd and Weisstein, Eric W. "Antihermitian Matrix." From MathWorld --A
Wolfram Web Resource. https://mathworld.wolfram.com/AntihermitianMatrix.html

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