unitary

unitary

0.1

Definitions

•

A

unitary space

V

is a

complex vector space

with a
distinguished

positive definite

Hermitian form

,

⟨

-

,

-

⟩

:

V

×

V

→

ℂ

,

which serves as
the

inner product

on

V

.

•

A

unitary transformation

is a

surjective

linear transformation

T

:

V

→

V

satisfying

⟨

u

,

v

⟩

=

⟨

T

⁢

u

,

T

⁢

v

⟩

,

u

,

v

∈

V

.

(1)

These are

isometries

of

V

.

•

More generally, a

unitary transformation

is a surjective linear transformation

T

:

U

⟶

V

between two unitary spaces

U

,

V

satisfying

⟨

T

⁢

v

,

T

⁢

u

⟩

V

=

⟨

v

,

u

⟩

U

,

u

,

v

∈

U

In this entry will restrict to the case of the first , i.e.

U

=

V

.

•

A

unitary matrix

is a square complex-valued matrix,

A

, whose

inverse

is equal to its

conjugate transpose

:

A

-

1

=

A

¯

t

.

•

When

V

is a

Hilbert space

, a

bounded linear operator

T

:

V

⟶

V

is said to be a

unitary operator

if its inverse is equal to its

adjoint

:

T

-

1

=

T

*

In Hilbert spaces unitary transformations correspond precisely to unitary operators.

0.2

Remarks

1.

A standard example of a unitary space is

ℂ

n

with inner product

⟨

u

,

v

⟩

=

∑

i

=

1

n

u

i

⁢

v

i

¯

,

u

,

v

∈

ℂ

n

.

(2)

2.

Unitary transformations and unitary matrices
are closely related. On the one hand, a unitary matrix defines a
unitary transformation of

ℂ

n

relative to the inner product
(

2

). On the other hand, the representing matrix of a
unitary transformation relative to an

orthonormal basis

is, in fact, a
unitary matrix.

3.

A unitary transformation is an

automorphism

. This follows from
the fact that a unitary transformation

T

preserves

the
inner-product norm:

∥

T

⁢

u

∥

=

∥

u

∥

,

u

∈

V

.

(3)

Hence, if

T

⁢

u

=

0

,

then by the

definition

(

1

)
it follows that

∥

u

∥

=

0

,

and hence by the inner-product axioms that

u

=

0

.

Thus, the kernel of

T

is

trivial

, and therefore it is an
automorphism.

4.

Moreover,

relation

(

3

) can be taken as the definition
of a unitary transformation. Indeed, using the

polarization
identity

it is possible to show that if

T

preserves the norm, then (

1

) must hold as well.

5.

A simple example of a unitary matrix is the

change of
coordinates

matrix between two orthonormal bases. Indeed, let

u

1

,

…

,

u

n

and

v

1

,

…

,

v

n

be two orthonormal bases, and
let

A

=

(

A

j

i

)

be the corresponding change of basis matrix
defined by

v

j

=

∑

i

A

j

i

⁢

u

i

,

j

=

1

,

…

,

n

.

Substituting the above relation into the

defining relations

for an
orthonormal basis,

⟨

u

i

,

u

j

⟩

=

δ

i

⁢

j

,

⟨

v

k

,

v

l

⟩

=

δ

k

⁢

l

,

we obtain

∑

i

⁢

j

δ

i

⁢

j

⁢

A

k

i

⁢

A

l

j

¯

=

∑

i

A

k

i

⁢

A

l

i

¯

=

δ

k

⁢

l

.

In matrix notation, the above is simply

A

⁢

A

¯

t

=

I

,

as desired.

6.

Unitary transformations form a group under

composition

. Indeed, if

S

,

T

are unitary transformations then

S

⁢

T

is also surjective and

⟨

S

⁢

T

⁢

u

,

S

⁢

T

⁢

v

⟩

=

⟨

T

⁢

u

,

T

⁢

v

⟩

=

⟨

u

,

v

⟩

for every

u

,

v

∈

V

. Hence

S

⁢

T

is also a unitary transformation.

7.

Unitary spaces,

transformations

, matrices and

operators

are of fundamental
importance in quantum mechanics.

Title

unitary

Canonical

name

Unitary

Date of creation

2013-03-22 12:02:01

Last modified on

2013-03-22 12:02:01

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

21

Author

asteroid (17536)

Entry type

Definition

Classification

msc 47D03

Classification

msc 47B99

Classification

msc 47A05

Classification

msc 46C05

Classification

msc 15-00

Synonym

complex inner product space

Related topic

EuclideanVectorSpace2

Related topic

PauliMatrices

Defines

unitary space

Defines

unitary matrix

Defines

unitary transformation

Defines

unitary operator

Defines

unitary group

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