Rank reduction

Step 3. Dimension (rank) reduction by only retaining A major components to approximate Y. This gives the RRR fit ... 

Figure 3. A diagrammatic representation of the cumulant decomposition ([IV, A2]p 2)) for the three-particle operator drawn in Fig. 2. Four kinds of one- and two-particle operators are obtained. The double line is the contraction for the particle-rank reduction (closure), where the correlation is averaged with the effective field (i.e., density matrices).

Eq. (120) is the same commutation relation as is satisfied by the generators of the unitary group U(n), the group of all n-dimensional unitary matrices. For this reason, the operators are often referred to as generators " . The operator rank reductions shown in Eqs (120) and (121) prove to be important in some of the following derivations. 

Not every restriction on the data gives a rank reduction. Consider again an example where spectroscopic measurements are performed on different samples. Suppose that there are three absorbing species in the samples with pure spectra si, S2 and S3, and concentration ci, c2 and c3, indicating the concentrations of the different absorbing species in the samples. A restriction often encountered is closure, which means that ci + c2 + c3 = 1, where 1 is a vector of ones.2 Closure occurs, for instance, if the three species are the absorbing species of a batch process because closure is then forced by the law of preservation of mass. 

In this appendix it is shown that the combination of closure and column centering gives rank reduction in certain cases. 

The basic theory behind the generalized rank annihilation method is that the rank reduction can be re-expressed and automated. A scalar y (relative concentration of the analyte in the unknown sample) is sought such that the matrix pencil... 

There is one value of y that will lead to a rank-reduction of the expression in Equation (6.58) namely the one for which y equals the concentration of the analyte in X2 relative to the concentration in Xi. It is assumed that Xi and X2 are square (R x R) matrices. Then seeking a scalar y with the property that the rank of X2 — y Xi is R — 1 can be formalized in... 

Hence, in both cases the rank of X is reduced by one. The reverse also holds if for the noiseless case no rank reduction of X is obtained upon column-centering, then the above model is not valid. To summarize, in the noiseless case there is a simple relation between the validity of above model and rank reduction upon column-centering. 

Similarly to our treatment of the one-electron contribution, we can commute operators to achieve a rank reduction through cancellation of two terms, and we are left with... 

The existence of rigid-body and kinematic modes is indicated by a rank reduction of the matrix G. Two solutions are proposed for solving this problem to perform the eigenvalue analysis in the presence of singularities. 

Before considering the evaluation and simplihcadon of commutators and anticommutators, it is useful to introduce the concepts of operator rank and rank reduction. The (particle) rank of a string of creation and annihilation operators is simply the number of elementary operators divided by 2. For example, Ihe rank of a creation operator is 1/2 and the rank of an ON operator is 1. Rank reduction is said to occur when the rank of a commutator or anticommutator is lower than the combined rank of the operators commuted or anticommuted. Consider the basic anticommutation relation... 

Note that the commutators and anticommutators on the right-hand side of these expressions contain fewer operators than does the commutator or the anticommutator on the left-hand side. For proofs, see Exercise 1.3. In deciding what identity to apply in a given case, we follow the principle of rank reduction - that is, we try to expand the expression in commutators or anticommutators that, to the greatest extent possible, exhibit rank reduction. 

Let us finally consider nested commutators. Nested commutators may be simplified by the same techniques as the single commutators, thus giving rise to rank reductions greater than 1. For example, the following double commutator is easily evaluated using (1.8.14) ... 

Rank reduction occurs in all cases since the singlet excitation operator is a rank-1 operator. These commutators are proved in Exercise 2.3. 

This result holds also when Op and Ob are linear combinations of strings (each of the same rank) and agrees with the rank-reduction rule given in Section 1.8, which states that rank reduction occurs upon anticommutation of two operators of half-integral rank and upon commutation of all... 

Note that the second and third steps hold for closed-shell states only and that the last step holds for real wave functions only. From the rules for rank reduction in Section 1.8, we also note that the anticommutator W]]+ in (10.3.8) is a one-electron operator. 

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