Skew-Hermitian matrix explained
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-Hermitianbecause
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.
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
.
is skew-Hermitian, then the
matrix exponential
is
unitary.
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
- , §4.1.1;, §3.2
- , §4.1.2
- , §2.5.2, §2.5.4
- , Exercise 3.2.5
- , §4.1.1
References