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. That is, the matrix A {\displaystyle A} is skew-Hermitian if it satisfies the relation
where A H {\displaystyle A^{\textsf {H}}} denotes the conjugate transpose of the matrix A {\displaystyle A} . In component form, this means that
for all indices i {\displaystyle i} and j {\displaystyle j} , where a i j {\displaystyle a_{ij}} is the element in the i {\displaystyle i} -th row and j {\displaystyle j} -th column of A {\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. The set of all skew-Hermitian n × n {\displaystyle n\times n} matrices forms the u ( n ) {\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 Hermitian product. Note that the adjoint of an operator depends on the scalar product considered on the n {\displaystyle n} dimensional complex or real space K n {\displaystyle K^{n}} . If ( ⋅ ∣ ⋅ ) {\displaystyle (\cdot \mid \cdot )} denotes the scalar product on K n {\displaystyle K^{n}} , then saying A {\displaystyle A} is skew-adjoint means that for all u , v ∈ K n {\displaystyle \mathbf {u} ,\mathbf {v} \in K^{n}} one has ( A u ∣ v ) = − ( u ∣ A v ) {\displaystyle (A\mathbf {u} \mid \mathbf {v} )=-(\mathbf {u} \mid A\mathbf {v} )} . In dimension 1, the skew-adjoint operators are exactly the imaginary numbers, whereas real numbers correspond to self-adjoint operators.
Example For example, the following matrix is skew-Hermitian
A = [ − i 2 + i − 2 + i 0 ] {\displaystyle A={\begin{bmatrix}-i&2+i\\-2+i&0\end{bmatrix}}}
because
… excerpt ends here. Continue reading the full article.
