Since the eigenvalues of a Hermitian matrix are real, we see that the eigenval
ues of a skewHermitian matrix are imaginary.
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Appendix A
Fields and vector spaces
Fields
A
field
is an algebraic structure
K
in which we can add and multiply elements,
such that the following laws hold:
Addition laws
(FA0) For any
a
,
b
∈
K
, there is a unique element
a
+
b
∈
K
.
(FA1) For all
a
,
b
,
c
∈
K
, we have
a
+(
b
+
c
) = (
a
+
b
)+
c
.
(FA2) There is an element 0
∈
K
such that
a
+
0
=
0
+
a
=
a
for all
a
∈
K
.
(FA3) For any
a
∈
K
, there exists

a
∈
K
such that
a
+(

a
) = (

a
)+
a
=
0.
(FA4) For any
a
,
b
∈
K
, we have
a
+
b
=
b
+
a
.
Multiplication laws
(FM0) For any
a
,
b
∈
K
, there is a unique element
ab
∈
K
.
(FM1) For all
a
,
b
,
c
∈
K
, we have
a
(
bc
) = (
ab
)
c
.
(FM2) There is an element 1
∈
K
, not equal to the element 0 from (FA2), such
that
a
1
=
1
a
=
a
for all
a
∈
K
.
(FM3) For any
a
∈
K
with
a
=
0, there exists
a

1
∈
K
such that
aa

1
=
a

1
a
=
1.
(FM4) For any
a
,
b
∈
K
, we have
ab
=
ba
.
Distributive law
(D) For all
a
,
b
,
c
∈
K
, we have
a
(
b
+
c
) =
ab
+
ac
.
89
90
APPENDIX A. FIELDS AND VECTOR SPACES
Note the similarity of the addition and multiplication laws. We say that
(
K
,
+)
is an
abelian group
if (FA0)–(FA4) hold. Then (FM0)–(FM4) say that
(
K
\{
0
}
,
·
)
is also an abelian group. (We have to leave out 0 because, as (FM3) says, 0 does
not have a multiplicative inverse.)
Examples of fields include
Q
(the rational numbers),
R
(the real numbers),
C
(the complex numbers), and
F
p
(the integers mod
p
, for
p
a prime number).
Associated with any field
K
there is a nonnegative integer called its
character
istic
, defined as follows. If there is a positive integer
n
such that 1
+
1
+
···
+
1
=
0,
where there are
n
ones in the sum, then the smallest such
n
is prime. (For if
n
=
rs
,
with
r
,
s
>
1, and we denote the sum of
n
ones by
n
·
1, then
0
=
n
·
1
= (
r
·
1
)(
s
·
1
)
;
by minimality of
n
, neither of the factors
r
·
1 and
s
·
1 is zero. But in a field, the
product of two nonzero elements is nonzero.) If so, then this prime number is
the characteristic of
K
. If no such
n
exists, we say that the characteristic of
K
is
zero.
For our important examples,
Q
,
R
and
C
all have characteristic zero, while
F
p
has characteristic
p
.
Vector spaces
Let
K
be a field. A
vector space V
over
K
is an algebraic structure in which we
can add two elements of
V
, and multiply an element of
V
by an element of
K
(this
is called
scalar multiplication
), such that the following rules hold:
Addition laws
(VA0) For any
u
,
v
∈
V
, there is a unique element
u
+
v
∈
V
.
(VA1) For all
u
,
v
,
w
∈
V
, we have
u
+(
v
+
w
) = (
u
+
v
)+
w
.
(VA2) There is an element 0
∈
V
such that
v
+
0
=
0
+
v
=
av
for all
v
∈
V
.
(VA3) For any
v
∈
V
, there exists

v
∈
V
such that
v
+(

v
) = (

v
)+
v
=
0.
(VA4) For any
u
,
v
∈
V
, we have
u
+
v
=
v
+
u
.
Scalar multiplication laws
(VM0) For any
a
∈
K
,
v
∈
V
, there is a unique element
av
∈
V
.
(VM1) For any
a
∈
K
,
u
,
v
∈
V
, we have
a
(
u
+
v
) =
au
+
av
.