complexification of vector space

complexification of vector space

0.1

Complexification of vector space

If

V

is a

real vector space

,
its

complexification

V

ℂ

is the

complex vector space

consisting of

elements

x

+

i

⁢

y

, where

x

,

y

∈

V

.

Vector addition

and

scalar multiplication

by

complex numbers

are defined in the obvious way:

(

x

+

i

⁢

y

)

+

(

u

+

i

⁢

v

)

=

(

x

+

u

)

+

i

⁢

(

y

+

v

)

,

x

,

y

,

u

,

v

∈

V

(

α

+

i

⁢

β

)

⁢

(

x

+

i

⁢

y

)

=

(

α

⁢

x

-

β

⁢

y

)

+

i

⁢

(

β

⁢

x

+

α

⁢

y

)

,

x

,

y

∈

V

,

α

,

β

∈

ℝ

.

If

v

1

,

…

,

v

n

is a basis for

V

, then

v

1

+

i

⁢

0

,

…

,

v

n

+

i

⁢

0

is a basis for

V

ℂ

. Naturally,

x

+

i

⁢

0

∈

V

ℂ

is often written just as

x

.

So, for example, the complexification of

ℝ

n

is (

isomorphic

to)

ℂ

n

.

0.2

Complexification of linear transformation

If

T

:

V

→

W

is a

linear transformation

between two real vector spaces

V

and

W

,
its complexification

T

ℂ

:

V

ℂ

→

W

ℂ

is defined by

T

ℂ

⁢

(

x

+

i

⁢

y

)

=

T

⁢

x

+

i

⁢

T

⁢

y

.

It may be readily verified that

T

ℂ

is complex-linear.

If

v

1

,

…

,

v

n

is a basis for

V

,

w

1

,

…

,

w

m

is a basis for

W

,
and

A

is the

matrix representation

of

T

with respect to these bases,
then

A

, regarded as a

complex matrix

,
is also the

representation

of

T

ℂ

with respect to the corresponding bases
in

V

ℂ

and

W

ℂ

.

So, the complexification process is a formal, coordinate-free way of saying:
take the matrix

A

of

T

, with its real entries,
but operate on it as a complex matrix. The advantage of making this abstracted definition is that
we are not required to fix a choice of

coordinates

and use matrix representations
when otherwise there is no need to.
For example, we might want to make

arguments

about the complex eigenvalues
and eigenvectors for a

transformation

T

:

V

→

V

, while,
of course, non-real eigenvalues and eigenvectors, by definition, cannot exist
for a transformation between real vector spaces.
What we really mean are the eigenvalues and eigenvectors of

T

ℂ

.

Also, the complexification process generalizes without change for infinite-dimensional
spaces.

0.3

Complexification of inner product

Finally,
if

V

is also a real

inner product space

,
its real

inner product

can be extended to a complex inner product for

V

ℂ

by
the obvious expansion:

⟨

x

+

i

⁢

y

,

u

+

i

⁢

v

⟩

=

⟨

x

,

u

⟩

+

⟨

y

,

v

⟩

+

i

⁢

(

⟨

y

,

u

⟩

-

⟨

x

,

v

⟩

)

.

It follows that

∥

x

+

i

⁢

y

∥

2

=

∥

x

∥

2

+

∥

y

∥

2

.

0.4

Complexification of norm

More generally, for a real

normed vector space

V

,
the

equation

∥

x

+

i

⁢

y

∥

2

=

∥

x

∥

2

+

∥

y

∥

2

can serve as a definition of the norm for

V

ℂ

.

References

1

Vladimir I. Arnol’d.

Ordinary Differential Equations

. Springer-Verlag, 1992.

Title

complexification of vector space

Canonical name

ComplexificationOfVectorSpace

Date of creation

2013-03-22 15:24:33

Last modified on

2013-03-22 15:24:33

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

9

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 15A04

Classification

msc 15A03

Related topic

ComplexStructure2

Defines

complexification

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