 TOPICS  # Antihermitian Matrix

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.

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