Θ Theta Criteria

Multivariate Analysis

By Steve Halitsky and Edward Halitsky
Theta Criteria Slide: Formal Study

II: Theta Criteria Formal Study

The Theta Criteria methods for positively defined Theta Criteria matrices were introduced in [1] - [4]. Those criteria have been constructed on norms of differences of matrices ordered weighted eigenvectors. We will now study Theta Criteria properties in depth. 

2.1.   Formal Definitions

Let Theta Criteriais real numbers field, Theta Criteria-finite linear vector space over Theta Criteria, Theta Criteria andTheta Criteria-set of all positively defined matrices of order Theta Criteria. Let block matrices Theta Criteria,          Theta Criteria, Theta Criteria                                                   (1)

sub matrices Theta Criteria, Theta Criteria , Theta Criteria and Theta Criteria, Theta Criteria , Theta Criteria.

Let   Theta Criteria - sets of all ordered eigenvalues of Theta Criteria:

Theta Criteria,                               (2)

andTheta Criteria and Theta Criteria sets of all Theta Criteriaorthonormalized eigenvectors.

Let Theta Criteria - eigenpair of Theta Criteria and Theta Criteriaand Theta Criteria, with Theta Criteria Theta Criteria - a set of pairs of eigenpairs of Theta Criteria-th eigenvalues and eigenvectors of Theta Criteria.

Let Theta Criteria, Theta Criteria  be a set of two pairs of eigenpairs of Theta Criteria-th and j-th eigenvalues and eigenvectors of Theta Criteria and Theta Criteria has been composed on Theta Criteria eigenpairs Theta Criteria.       

2.2.    Known Matrices Closeness Criteria

The forward differences Theta Criteria of the determinants and condition numbers were used as matrices closeness criteria [5] - [9]:  

Theta Criteria                                   (3)

Theta Criteria                               (4)

The Theta Criteria criteria of Theta Criteria has been introduced in [4]: Theta Criteria.                                                                            (5)

2.3.    The Hilbert Theorem

Let Theta Criteriabe a limited linear, self-conjugated integral matrix from space Theta Criteria into Theta Criteriaand 

Theta CriteriawhereTheta Criteria is the matrix's kernel. There exist Theta Criteria representations on orthonormalized matrices of eigenfunctions Theta Criteriaand eigenvalues Theta Criteria of Theta Criteriaas follows:

            Theta Criteria                                  (6)

Theta Criteria(7)

The series are converging on norms Theta Criteria respectively [10] - [12].

2.4.    Theta Criteria or Θ Criteria Construction

Let us construct Theta Criteriacriteria between Theta Criteria, or Theta Criteria, which can converge on Theta Criteria. Such criteria will reflect the geometrical changes on some the elements of Theta Criteria, Theta Criteriaor Theta Criteria. The proper choice of Theta Criteriacriteria depends on a priori information about Theta Criteriastructures and their distinction type. If all elements of Theta Criteria have changed, then Theta Criteriais appropriate choice. If only Theta Criteriaand Theta Criteriahave changed, then Theta Criteria is acceptable. Now we can formulate several hypotheses about matrices Theta Criteriadifferences.

Hypothesis I: The matrices Theta Criteriadistinctions can be represented by geometrical differences between Theta Criteriaand Theta Criteria of Theta Criteria.

Then Euclidean normTheta Criteria ofTheta Criteria can serve as Theta Criteria or

Theta Criteria.(8)

Hypothesis II: The matrices Theta Criteria distinction is represented by geometrical differences between Theta Criteriaand Theta Criteriaof Theta Criteria.

Then the sum of Theta Criteria and Theta Criteria can serve as Theta Criteria:

Theta Criteria=Theta Criteria   (9)

Hypothesis III: The matrices Theta Criteria distinction is represented by geometrical differences between Theta Criteriaof Theta Criteria.

Then the sum of Theta Criteria, Theta CriteriaTheta Criteria can serve as Theta CriteriaTheta Criteria.                                                                        (10)

According to [9], a real-valued function Theta Criteria on linear space Theta Criteria Theta Criteria is Theta Criteria on Theta Criteria, if

Theta Criteria(Positivity)                                        (11)Theta Criteria                      (Triangle inequality)                       (12)   Theta Criteria                               (Homogeneity) (13)

Theta Criteriaif and only if Theta Criteria.(Positive definiteness)                    (14)

2.5.    Theta Criteria Properties

Theorem 1. (Positivity).

The criteria Theta Criteria.

Proof: From Theta Criteriacriteria definition and Euclidean norm properties

Theta Criteria

Q.E.D.                                                                                  


Theorem 2.(Triangle inequality) 

If Theta CriteriaTheta Criteria, then Theta Criteria.   

Proof: According to Theta Criteriadefinition, Theta Criteria,Theta Criteria,Theta Criteria.

Since Theta Criteria, the vectors Theta Criteria. Then Theta Criteria.
Q.E.D.

Theorem 3.(Homogeneity):

Theta Criteria,whereTheta Criteria.                                                       

Proof: Since Theta Criteria,

Theta Criteria

Q.E.D.


Theorem 4. (Positive definiteness)

Theta Criteriaif and only ifTheta Criteria.                                                         

Proof: Let Theta Criteria.    The Theta Criteriacriteria is Theta Criteria. The Theta Criteria-th component of Theta Criteriais

Theta Criteria. According to [2], [3] and Hilbert Theorem Theta Criteria  and Theta Criteria.

Then Theta Criteria is true, because index Theta Criteria is arbitrary. 

Let Theta Criteria.                                         

Then we will receive the system of Theta Criteria equations

Theta Criteria

with solution Theta Criteria. According to the Hilbert theorem, for each Theta CriteriaandTheta Criteriaexist uniqueTheta Criteria and Theta Criteria.  If Theta Criteria, then Theta Criteria, Theta Criteriaand Theta Criteria.

Conclusion: The criteria Theta Criteriais a norm on Theta Criteria.

Theorem 5. (symmetry)

If  Theta Criteria and Theta Criteria thenTheta Criteria.                                                                                

Proof: If Theta CriteriaandTheta Criteriaswitch places in Theta Criteria, then Theta Criteria.

Theorem 6.

If Theta Criteria, Theta Criteria, then the Theta Criteria is the matrix norm difference

Theta Criteria

Proof:

Criteria Theta Criteria.

Theorem 7.

If Theta Criteria, Theta Criteria then Theta Criteria.                                                                                                

Proof:

FromTheta Criteria, and Theta Criteria we received:

Theta Criteria 

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