Skew-Hermitian matrix

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.^[1] That is, the matrix ${\displaystyle A}$ is skew-Hermitian if it satisfies the relation

where ${\displaystyle A^{\textsf {H}}}$ denotes the conjugate transpose of the matrix ${\displaystyle A}$. In component form, this means that

for all indices ${\displaystyle i}$ and ${\displaystyle j}$, where ${\displaystyle a_{ij}}$ is the element in the ${\displaystyle i}$-th row and ${\displaystyle j}$-th column of ${\displaystyle A}$, 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 ${\displaystyle n\times n}$ matrices forms the ${\displaystyle u(n)}$ 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 ${\displaystyle n}$ dimensional complex or real space ${\displaystyle K^{n}}$. If ${\displaystyle (\cdot \mid \cdot )}$ denotes the scalar product on ${\displaystyle K^{n}}$, then saying ${\displaystyle A}$ is skew-adjoint means that for all ${\displaystyle \mathbf {u} ,\mathbf {v} \in K^{n}}$ one has ${\displaystyle (A\mathbf {u} \mid \mathbf {v} )=-(\mathbf {u} \mid A\mathbf {v} )}$.

Imaginary numbers can be thought of as skew-adjoint (since they are like ${\displaystyle 1\times 1}$ matrices), whereas real numbers correspond to self-adjoint operators.