Matrix whose conjugate transpose is its negative (additive inverse)
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 is skew-Hermitian if it satisfies the relation
where denotes the conjugate transpose of the matrix . In component form, this means that
for all indices and , where 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 . If denotes the scalar product on , then saying is skew-adjoint means that for all one has .
Imaginary numbers can be thought of as skew-adjoint (since they are like matrices), whereas
real numbers correspond to
self-adjoint operators.
Example
For example, the following matrix is skew-Hermitian
because
Properties
- The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are
normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
[3]
- All entries on the
main diagonal of a skew-Hermitian matrix have to be pure
imaginary; i.e., on the imaginary axis (the number zero is also considered purely imaginary).
[4]
- If and are skew-Hermitian, then is skew-Hermitian for all
real
scalars and .
[5]
- is skew-Hermitian if and only if (or equivalently, ) is
Hermitian.
[5]
- is skew-Hermitian if and only if the real part is
skew-symmetric and the imaginary part is
symmetric.
- If is skew-Hermitian, then is Hermitian if is an even integer and skew-Hermitian if is an odd integer.
- is skew-Hermitian if and only if for all vectors .
- If is skew-Hermitian, then the
matrix exponential is
unitary.
- The space of skew-Hermitian matrices forms the
Lie algebra of the
Lie group .
Decomposition into Hermitian and skew-Hermitian
- The sum of a square matrix and its conjugate transpose is Hermitian.
- The difference of a square matrix and its conjugate transpose is skew-Hermitian. This implies that the
commutator of two Hermitian matrices is skew-Hermitian.
- An arbitrary square matrix can be written as the sum of a Hermitian matrix and a skew-Hermitian matrix :
See also
Notes
References