In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix.
is skew-Hermitian if it satisfies the relation
denotes the conjugate transpose of the matrix
In component form, this means that
{\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad a_{ij}=-{\overline {a_{ji}}}}
for all indices
{\displaystyle a_{ij}}
is the element in the
-th row and
-th column of
, and the overline denotes complex conjugation.
Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.
[2] The set of all skew-Hermitian
matrices forms the
Lie algebra, which corresponds to the Lie group U(n).
The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.
Note that the adjoint of an operator depends on the scalar product considered on the
dimensional complex or real space
denotes the scalar product on
is skew-adjoint means that for all
Imaginary numbers can be thought of as skew-adjoint (since they are like
matrices), whereas real numbers correspond to self-adjoint operators.
For example, the following matrix is skew-Hermitian
{\displaystyle A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix}}}
{\displaystyle -A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {-2+i}}\\{\overline {2+i}}&{\overline {0}}\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {2+i}}\\{\overline {-2+i}}&{\overline {0}}\end{bmatrix}}^{\mathsf {T}}=A^{\mathsf {H}}}