An
n
by
n
complex or real matrix
A
=
(
a
i
,
j
)
1
≤
i
,
j
≤
n
is said to be anti-Hermitian, skew-Hermitian, or said to represent a skew-adjoint operator, or to be a skew-adjoint matrix, on the complex or real
n
dimensional space
K
n
, if its adjoint is the negative of itself: :
A
∗
=
−
A
.
Note that the adjoint of an operator depends on the scalar product considered on the
n
dimensional complex or real space
K
n
. If
(
⋅
|
⋅
)
denotes the scalar product on
K
n
, then saying
A
is skew-adjoint means that for all
u
,
v
∈
K
n
one has
(
A
u
|
v
)
=
−
(
u
|
A
v
)
.
In the particular case of the canonical scalar products on
K
n
, the matrix of a skew-adjoint operator satisfies
a
i
j
=
−
a
¯
j
i
for all
1
≤
i
,
j
≤
n
.
Imaginary numbers can be thought of as skew-adjoint (since they are like 1-by-1 matrices), whereas real numbers correspond to self-adjoint operators.
