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.
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 .
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
realscalars 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 .
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 :