- Source: Skew-Hermitian matrix
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. That is, the matrix
A
{\displaystyle A}
is skew-Hermitian if it satisfies the relation
where
A
H
{\displaystyle A^{\textsf {H}}}
denotes the conjugate transpose of the matrix
A
{\displaystyle A}
. In component form, this means that
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 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. The set of all skew-Hermitian
n
×
n
{\displaystyle n\times n}
matrices forms the
u
(
n
)
{\displaystyle u(n)}
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
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} )}
.
Imaginary numbers can be thought of as skew-adjoint (since they are like
1
×
1
{\displaystyle 1\times 1}
matrices), whereas real numbers correspond to self-adjoint operators.
Example
For example, the following matrix is skew-Hermitian
A
=
[
−
i
+
2
+
i
−
2
+
i
0
]
{\displaystyle A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix}}}
because
−
A
=
[
i
−
2
−
i
2
−
i
0
]
=
[
−
i
¯
−
2
+
i
¯
2
+
i
¯
0
¯
]
=
[
−
i
¯
2
+
i
¯
−
2
+
i
¯
0
¯
]
T
=
A
H
{\displaystyle -A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {-2+i}}\\{\overline {2+i}}&{\overline {0}}\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {2+i}}\\{\overline {-2+i}}&{\overline {0}}\end{bmatrix}}^{\mathsf {T}}=A^{\mathsf {H}}}
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.
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).
If
A
{\displaystyle A}
and
B
{\displaystyle B}
are skew-Hermitian, then
a
A
+
b
B
{\displaystyle aA+bB}
is skew-Hermitian for all real scalars
a
{\displaystyle a}
and
b
{\displaystyle b}
.
A
{\displaystyle A}
is skew-Hermitian if and only if
i
A
{\displaystyle iA}
(or equivalently,
−
i
A
{\displaystyle -iA}
) is Hermitian.
A
{\displaystyle A}
is skew-Hermitian if and only if the real part
ℜ
(
A
)
{\displaystyle \Re {(A)}}
is skew-symmetric and the imaginary part
ℑ
(
A
)
{\displaystyle \Im {(A)}}
is symmetric.
If
A
{\displaystyle A}
is skew-Hermitian, then
A
k
{\displaystyle A^{k}}
is Hermitian if
k
{\displaystyle k}
is an even integer and skew-Hermitian if
k
{\displaystyle k}
is an odd integer.
A
{\displaystyle A}
is skew-Hermitian if and only if
x
H
A
y
=
−
y
H
A
x
¯
{\displaystyle \mathbf {x} ^{\mathsf {H}}A\mathbf {y} =-{\overline {\mathbf {y} ^{\mathsf {H}}A\mathbf {x} }}}
for all vectors
x
,
y
{\displaystyle \mathbf {x} ,\mathbf {y} }
.
If
A
{\displaystyle A}
is skew-Hermitian, then the matrix exponential
e
A
{\displaystyle e^{A}}
is unitary.
The space of skew-Hermitian matrices forms the Lie algebra
u
(
n
)
{\displaystyle u(n)}
of the Lie group
U
(
n
)
{\displaystyle U(n)}
.
Decomposition into Hermitian and skew-Hermitian
The sum of a square matrix and its conjugate transpose
(
A
+
A
H
)
{\displaystyle \left(A+A^{\mathsf {H}}\right)}
is Hermitian.
The difference of a square matrix and its conjugate transpose
(
A
−
A
H
)
{\displaystyle \left(A-A^{\mathsf {H}}\right)}
is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
An arbitrary square matrix
C
{\displaystyle C}
can be written as the sum of a Hermitian matrix
A
{\displaystyle A}
and a skew-Hermitian matrix
B
{\displaystyle B}
:
C
=
A
+
B
with
A
=
1
2
(
C
+
C
H
)
and
B
=
1
2
(
C
−
C
H
)
{\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)}
See also
Bivector (complex)
Hermitian matrix
Normal matrix
Skew-symmetric matrix
Unitary matrix
Notes
References
Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6.
Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8.
Kata Kunci Pencarian:
- Matriks normal
- Transpos konjugat
- Skew-Hermitian matrix
- Hermitian matrix
- Skew-symmetric matrix
- Sesquilinear form
- Unitary matrix
- Skew
- Square matrix
- Transpose
- Symmetric matrix
- List of things named after Charles Hermite
