Skew-Hermitian matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> with <a href="/facts/Complex_number/2w7mqI7e">complex</a> entries is said to be skew-Hermitian or anti-Hermitian if its <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> is the negative of the original matrix. That is, the matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is skew-Hermitian if it satisfies the relation

A
 
  skew-Hermitian
 
 
 
 ⟺
 
 
 
 A
 
 
 H
 
 
 
 =
 −
 A
 
 
 {\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad A^{\mathsf {H}}=-A}

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

A
 
  skew-Hermitian
 
 
 
 ⟺
 
 
 
 a
 
 i
 j
 
 
 =
 −
 
 
 
 a
 
 j
 i
 
 
 ¯
 
 
 
 
 {\displaystyle A{\text{ skew-Hermitian}}\quad \iff \quad a_{ij}=-{\overline {a_{ji}}}}

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 <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugation</a>.
Skew-Hermitian matrices can be understood as the complex versions of real <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric matrices</a>, 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)}
 
 <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>, which corresponds to the Lie group <a href="/facts/Unitary_group/lEvudIhU">U(n)</a>. The concept can be generalized to include <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of any <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> with a <a href="/facts/Sesquilinear/Kp6qUA8v">sesquilinear</a> <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>.
Note that the <a href="/facts/Adjoint_operator/gN6RHphd">adjoint</a> of an operator depends on the <a href="/facts/Scalar_product/tNz8MLjT">scalar product</a> 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} )}
 
.
<a href="/facts/Imaginary_number/6Xh9W3hq">Imaginary numbers</a> can be thought of as skew-adjoint (since they are like 
 
 
 
 1
 ×
 1
 
 
 {\displaystyle 1\times 1}
 
 matrices), whereas <a href="/facts/Real_number/R02gw5Pb">real numbers</a> correspond to <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a> operators.

Skew-Hermitian matrix open-in-new

Skew-Hermitian matrix