II: Theta Criteria Formal Study

The Theta Criteria methods for positively defined  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 is real numbers field, -finite linear vector space over ,  and-set of all positively defined matrices of order . Let block matrices ,          ,                                                    (1)

sub matrices ,  ,  and ,  , .

Let    - sets of all ordered eigenvalues of :

,                               (2)

and and  sets of all orthonormalized eigenvectors.

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

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

2.2.    Known Matrices Closeness Criteria

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

                                   (3)

                               (4)

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

2.3.    The Hilbert Theorem

Let be a limited linear, self-conjugated integral matrix from space  into and 

where is the matrix's kernel. There exist  representations on orthonormalized matrices of eigenfunctions and eigenvalues  of as follows:

                                              (6)

(7)

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

2.4.    Theta Criteria or Θ Criteria Construction

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

Hypothesis I: The matrices distinctions can be represented by geometrical differences between and  of .

Then Euclidean norm of can serve as  or

.(8)

Hypothesis II: The matrices  distinction is represented by geometrical differences between and of .

Then the sum of  and  can serve as :

=   (9)

Hypothesis III: The matrices  distinction is represented by geometrical differences between of .

Then the sum of , … can serve as :  .                                                                        (10)

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

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

if and only if .(Positive definiteness)                    (14)

2.5.    Theta Criteria Properties

Theorem 1. (Positivity).

The criteria .

Proof: From criteria definition and Euclidean norm properties

Q.E.D.                                                                                  

Theorem 2.(Triangle inequality) 

If , then .   

Proof: According to definition, ,,.

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

Theorem 3.(Homogeneity):

,where.                                                       

Proof: Since ,

Q.E.D.

Theorem 4. (Positive definiteness)

if and only if.                                                         

Proof: Let .    The criteria is . The -th component of is

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

Then  is true, because index  is arbitrary. 

Let .                                         

Then we will receive the system of  equations

with solution . According to the Hilbert theorem, for each andexist unique and .  If , then , and .

Conclusion: The criteria is a norm on .

Theorem 5. (symmetry)

If   and  then.                                                                                

Proof: If andswitch places in , then .

Theorem 6.

If , , then the  is the matrix norm difference

Proof:

Criteria .

Theorem 7.

If ,  then .                                                                                                

Proof:

From, and  we received:

References:

1. V. Vasil'ev, V. Halitsky (same person as S. Halitsky / Steve Halitsky) and V. Revenko. Estimation of Correlations between Groups of Parameters of a Multidimensional Plant. Soviet Automat. Control. 1973, no. 2, 9-15.
2. V. Halitsky (same person as S. Halitsky / Steve Halitsky). Shortening of description of a dynamic control system with given constraints.
   The Proceedings of Ukrainian Academy of Science, Type A, 1985, #9, pp. 65-67.      
3. S. Halitsky. About closeness criteria of positively-defined operators. The Proceedings of 1st Bernoulli Congress on Mathematical Statistics, Vol. 2, pp. 926-927, "Nauka," Moscow, 1986.
4. S. Halitsky.  The closeness criteria for positively-defined operators      Express-Printing Edition, #97-4, pp. 1-17, 1997, Russian National Academy of Science.
5. H. Weyl. The Classical Groups, Their Invariants and Representations. Princeton, 1999.
6. G. W.  Stewart. Matrix algorithms. Vol. 1. Basic Decompositions. SIAM, 1998.
7. G. Golub, C. F. Van Loan. Matrix Computations. The John Hopkins Univ. Press, 3rd Ed., 1996.
8. N. J. Higham. Accuracy and Stability of Numerical Algorithms. SIAM, 1996.
9. V.B. Korotkov, Hilbert–Schmidt integral operator. Mathematics Encyclopedia, Vol. 1, p. 975. Soviet Encyclopedia, Moscow, 1977.
10. B.M. Bredichin, Hilbert–Schmidt integral operator. Encyclopaedia of Mathematics, Springer 2002.
11. N. Akhieser, I. Glazman. Theory of Linear Operators in Hilbert Space. Dover Ed., 1003.
12. R. Horn, C. Johnson. Matrix Analysis. Cambridge University Press. 1985.