square matrix is antihermitian if it satisfies
is the adjoint. For example, the matrix
is an antihermitian matrix. Antihermitian matrices are often called "skew Hermitian matrices" by mathematicians.
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
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.,
is the adjoint and is the identity matrix.
The derivative at of both sides must be equal so
That is, the
derivative of at the identity must be antihermitian.
exponential map of an antihermitian
matrix is a unitary matrix.
See also Adjoint
Portions of this entry contributed by
Rowland Explore with Wolfram|Alpha
Cite this as:
Rowland, Todd and Weisstein, Eric W. "Antihermitian Matrix." From --A
Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/AntihermitianMatrix.html