§ 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 β be vectors in unitary space, and c be a complex number, then

1 o

2 o    (Cauchy - Schwartz inequality)

The equal sign holds if and only if α and β are linearly related .

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 β . Obviously, if α is orthogonal to β , then β is also orthogonal to α .

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



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


(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 V on the field F , the transformation defined by the relational expression is a linear transformation , which is called the conjugate transformation of L. If , then L is called a normal transformation .

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 A under a certain standard orthonormal basis , then the matrix of conjugate transformation about the same basis is the conjugate transpose matrix of A.

    [ 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 M are self-conjugate transformations, they are also self-conjugate transformations . When L and M are commutative, LM is also self-conjugate transformation .

     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 L.

    [ Unitary transformation ] If   there is a linear transformation L for any in the unitary space V , then L is called a unitary transformation .

    The unitary transformation has the following properties:

    1 o The identity transformation is a unitary transformation .

    2 o If L and M are unitary transformations, then LM is also unitary transformations .

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:


    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 V , if there are two complementary subspaces S and T of V such that if , then


This transformation P is called the projection of V on S along T.

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 T on subspace S , then the matrix of P can be transformed into the following form:


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 V. Let be the different eigenvalues ​​of L , and let be the set of vectors α that make , then it is the subspace of V. Obviously , and is the orthonormal complement space of V. If { } is an orthonormal basis of Si , where is the dimension of , then the set { } composed of all these is a canonical basis of V. Finally , let P i is the projection of V on Si , then with respect to the base above, LThe matrix has the following form:


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 and any two vectors α , β The angle is defined as follows:


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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