**§ ****4 ****Unitary space**

1. Definition and properties of unitary space

[ Unitary space and Euclidean space ] Let *V* be a linear space on the complex number field *F* , if the inner product (quantity product) of two vectors is defined in *V* , denoted as ( ), and satisfy:_{}_{}

(i) ( ) = ( ), where ( ) is the complex conjugate of ( );_{}_{}_{}_{}

(ii) ( ) , the equal sign holds if and only then ;_{}_{}_{}

(iii) _{}, for any establishment;_{}_{}

*Then V* is called a unitary ( *U* ) space, also known as an inner product space .

If *F* is the field of real numbers, then the inner product is commutative . The finite-dimensional real unitary space is called Euclidean space .

For example, in an *n* -dimensional linear space , if specified _{}

_{}

in the formula

_{}

is a unitary space ._{}

The inner product in the unitary space *V* has the property:

1 ^{o} ( ) =_{}_{}

^{2o} _^{} _{}

3 ^{o} general, then_{}_{}

_{}

4o ^{_} _{}

[ modulus ( norm )] is real because of . Let_{}_{}

_{}

Call it the modulus or norm of a vector in the unitary space *V.* A vector whose modulus is 1 is called a unit vector or a standard vector ._{}

Let ** α** and

1 ^{o}_{}

2 ^{o}_{} (Cauchy - Schwartz inequality)

The equal sign holds if and only if ** α** and

^{3o} _^{}_{}

These properties are independent of the dimensionality of the space .

[ Orthogonal and Standard Orthogonal Basis ] In the unitary space *V* , if , the vector ** α** is said to be orthogonal to

In unitary space, any set of two orthogonal nonzero vectors is linearly independent .

If a set of unit vectors is orthogonal to each other, it is called a standard orthogonal set . If this vector set generates the entire space *V* , it is called the standard orthonormal basis of
*V.*

Let { } be _{}a set of standard orthonormal vectors in the unitary space *V* , , then_{}

1 ^{o}_{}
(Bessel's inequality)

2 ^{o}_{} is orthogonal to_{}

3 ^{o} When *V* is a finite-dimensional space, the necessary and sufficient condition for { } _{}to be the basis of *V* is: any vector can be expressed as_{}

_{}_{}

and _{}

[ Orthogonal Complementary Space of Subspace ] Let *V* be a unitary space on the complex field, and *S* be a subspace of *V , if*

(i)_{}

(ii) *T* is called the orthogonal complement space of *S* for the pair and there ._{}_{}_{}

It is immediately known from (i) (the empty set) ._{}

If *S* is a subspace of a finite-dimensional unitary space , then there is a subspace *T* that
is the orthogonal complement of *S.*_{}_{}

2. Special linear transformation on unitary space

[ Conjugate transformation ] For a linear transformation ** L** on the unitary space

Conjugate transformations have the following properties:

1 ^{o}_{}

^{2o} _^{}_{}

^{3o} _^{}_{}

4o ^{_}_{}

5 ^{o} If ** L** is a non-singular linear transformation, then it is also a non-singular linear transformation, and

_{}

6 ^{o If the matrix of }** L** is

[ Self-conjugate transformation (Hermitian transformation) ] If , then ** L** is called self-conjugate transformation or Hermitian transformation .

The self-conjugate transformation has the following properties:

1 ^{oIf }** L** and

2 ^{o} Under the standard orthonormal basis, the matrix of self-conjugate transformation is Hermitian matrix . On the contrary, if the matrix of linear transformation about a standard orthonormal basis is Hermitian matrix, it must be self-conjugate transformation .

3 ^{o} The eigenvalues of the autoconjugate transform are real .

4 ^{o} There are suitable standard orthonormal bases such that the self-conjugate transformation ** L** corresponds to a real diagonal matrix whose elements on the main diagonal are all the eigenvalues of

[ Unitary transformation ] If there is a linear transformation ** L** for any in the unitary space

The unitary transformation has the following properties:

1 ^{o} The identity transformation is a unitary transformation .

2 ^{o} If ** L** and

3 ^{o} If ** L** is a unitary transformation, it is also a unitary transformation .

The necessary and sufficient conditions for 4 ^{o }** L** to be a unitary transformation are:

_{} or _{}

5 ^{o} Under the standard orthonormal basis, the matrix of the unitary transformation ** L** is a unitary matrix .
On the contrary, if the matrix of the linear transformation about a standard orthonormal basis is a unitary matrix, it must be a unitary transformation .

^{}The absolute values of the eigenvalues of the 6o unitary transformation are ^{all} 1 .

3. Projection

[ Projection and its properties ] For a linear transformation ** P** on a linear space

_{}

This transformation ** P** is called the projection of

Projection has the following properties:

1 ^{o} If ** P** is a projective, then

_{}

Therefore, projection is an idempotent transformation; conversely, an idempotent transformation must be projective .

2 ^{o} If the linear space *V* is projective along and along respectively , then_{}_{}_{}_{}_{}

(i) is a projective if and only if if , then , and is a projective along the above ._{}_{}_{}_{}_{}_{}

(ii) If , then ** P** is a projective along .

3 ^{o} Let *T* and *S* be two complementary subspaces of a finite-dimensional linear space , and ** P** be the projection along subspace

_{}

where *A* is a square matrix of order *k* .

[ Orthoprojection ] Let *S* and *T* be the complementary subspace of the unitary space *V on the complex field, then the projection of **V* on *S* along *T* is called the orthoprojection of *V* on *S.*

[ Decomposition of self-conjugate transformation ] Let ** L** be a self-conjugate transformation on a finite -dimensional unitary space

_{} =_{}

where represents the order identity matrix . On the other hand, the matrix about this basis projection *P ** _{i}* is

_{}

where represents a zero matrix of order ._{}_{}

Therefore, the self-conjugate transformation can be written as a linear combination of projections .

_{}

4. Metrics in unitary space

In the first paragraph of this section, the modulus (norm) of each vector ** α** in the unitary space has been introduced . The distance between two "points" (ie vectors) α, β in the unitary space

_{}

The function defined by the above equation satisfies all conditions in the scale space (see Chapter 21, § 4 , a) .

If *V* is a real unitary space, then for everything , the angle must be real ._{}_{}

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