Antihermitian Matrix

A square matrix A is antihermitian if it satisfies


where A^(H) is the adjoint. For example, the matrix

 [i 1+i 2i; -1+i 5i 3; 2i -3 0]

is an antihermitian matrix. Antihermitian matrices are often called "skew Hermitian matrices" by mathematicians.

A matrix m can be tested to see if it is antihermitian in the Wolfram Language using AntihermitianMatrixQ[m].

The set of n×n antihermitian matrices is a vector space, and the commutator


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 A(t) is a path of unitary matrices through A(0)=I, i.e.,


for all t, where A^(H) is the adjoint and I is the identity matrix. The derivative at t=0 of both sides must be equal so


That is, the derivative of A(t) 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.

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