Skew-Hermitian matrix

All skew-Hermitian matrices are diagonalizable
In linear algebra, a square matrix with complex entries is said to be **skew-Hermitian** or **antihermitian** 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

A
 skew-Hermitian

⟺

A

H

=
−
A

{\displaystyle A \text{ skew-Hermitian} \quad \iff \quad A^\mathsf{H} = -A}

where

A

H

{\displaystyle A^{\text{H}}}

 denotes the conjugate transpose of the matrix

A

{\displaystyle A}

. In component form, this means that

A
 skew-Hermitian

⟺

a

i
j

=
−

a

j
i

¯

{\displaystyle A \text{ skew-Hermitian} \quad \iff \quad a\_{ij} = -\overline{a\_{ji}}}

for all indices

i

{\displaystyle i}

 and

j

{\displaystyle j}

, where

a

i
j

{\displaystyle a\_{ij}}

 is the element in the

j

{\displaystyle j}

-th row and

i

{\displaystyle i}

-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|\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 u,v \in K^n}

 one has

(
A
u

v
# )
−
(
u

A
v
)

{\displaystyle (Au|v) = - (u|Av) \,}

.

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 i A}

 (or equivalently,

−
i
A

{\displaystyle -i A}

) 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 x^\mathsf{H} A y = -y^\mathsf{H} A x}

 for all vectors

x
,
y

{\displaystyle x, 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)}