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
: ![{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0919d2e50fe1008af261f8301f243c002c328dbf)
See also
Notes
References