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

A

is skew-Hermitian if it satisfies the relation

where

Asf{H}

denotes the conjugate transpose of the matrix

A

. In component form, this means that

for all indices

i

and

j

, where

aij

is the element in the

i

-th row and

j

-th column of

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.[2] The set of all skew-Hermitian

n x n

matrices forms the

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

dimensional complex or real space

Kn

. If

(\mid)

denotes the scalar product on

Kn

, then saying

A

is skew-adjoint means that for all

u,v\inKn

one has

(Au\midv)=-(u\midAv)

.

Imaginary numbers can be thought of as skew-adjoint (since they are like

1 x 1

matrices), whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian A = \begin -i & +2 + i \\ -2 + i & 0 \endbecause -A = \begin i & -2 - i \\ 2 - i & 0 \end = \begin \overline & \overline \\ \overline & \overline \end = \begin \overline & \overline \\ \overline & \overline \end^\mathsf = A^\mathsf

Properties

A

and

B

are skew-Hermitian, then is skew-Hermitian for all real scalars

a

and

b

.[5]

A

is skew-Hermitian if and only if

iA

(or equivalently,

-iA

) is Hermitian.[5]

A

is skew-Hermitian if and only if the real part

\Re{(A)}

is skew-symmetric and the imaginary part

\Im{(A)}

is symmetric.

A

is skew-Hermitian, then

Ak

is Hermitian if

k

is an even integer and skew-Hermitian if

k

is an odd integer.

A

is skew-Hermitian if and only if

xHAy=-\overline{yHAx

} for all vectors

x,y

.

A

is skew-Hermitian, then the matrix exponential

eA

is unitary.

u(n)

of the Lie group

U(n)

.

Decomposition into Hermitian and skew-Hermitian

\left(A+AH\right)

is Hermitian.

\left(A-AH\right)

is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.

C

can be written as the sum of a Hermitian matrix

A

and a skew-Hermitian matrix

B

: C = A + B \quad\mbox\quad A = \frac\left(C + C^\mathsf\right) \quad\mbox\quad B = \frac\left(C - C^\mathsf\right)

See also

Notes

  1. , §4.1.1;, §3.2
  2. , §4.1.2
  3. , §2.5.2, §2.5.4
  4. , Exercise 3.2.5
  5. , §4.1.1

References