The Theta Criteria methods for positively defined matrices were introduced in  - . Those criteria have been constructed on norms of differences of matrices ordered weighted eigenvectors. We will now study Theta Criteria properties in depth.
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 :
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 .
The forward differences of the determinants and condition numbers were used as matrices closeness criteria  - :
The criteria of has been introduced in : . (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:
The series are converging on norms respectively  - .
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
Hypothesis II: The matrices distinction is represented by geometrical differences between and of .
Then the sum of and can serve as :
Hypothesis III: The matrices distinction is represented by geometrical differences between of .
Then the sum of , … can serve as : . (10)
According to , a real-valued function on linear space is on , if
(Positivity) (11) (Triangle inequality) (12) (Homogeneity) (13)
if and only if .(Positive definiteness) (14)
Theorem 1. (Positivity).
The criteria .
Proof: From criteria definition and Euclidean norm properties
Theorem 2.(Triangle inequality)
If , then .
Proof: According to definition, ,,.
Since , the vectors . Then .
Proof: Since ,
Theorem 4. (Positive definiteness)
if and only if.
Proof: Let . The criteria is . The -th component of is
. According to ,  and Hilbert Theorem and .
Then is true, because index is arbitrary.
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 .
If , , then the is the matrix norm difference
If , then .
From, and we received: