The trace of a matrix product

Consider two quadratic n x n matrices A and B with elements Oij and respectively. The trace of their product is defined as the sum of the diagonal components of matrix C = AB that is, 

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Now comes the very principle of the principal component analysis. A total variance is now defined as the trace of the matrix Sx or, using a property of the trace of a matrix product given in Section 2.2... 

Any mean value, which is the trace of a product of an operator matrix A, say) and a density matrix in the usual orbital basis representation becomes a simple scalar product in the basis-product representation ... 

A linear scaling algorithm for evaluation of three-body terms in the BH expansion has been described by Austin et al. [167, 168] Rewriting each term as a trace over a matrix product... 

Note that (4.7.15) cannot be factored into a product of integrals. This could have been done if G = 0 in (4.7.10). Nor can we write it as the trace of a 2 x 2 matrix as we have done in the simple Ising model. The situation is more complicated because of the hydrogen-bonding possibility included in t/(0, /). 

Tr designating the trace of the matrix product that follows. This is seen on direct expansion. Also, if a is a statistical matrix belonging to a certain sharp value of a specified observable, the probability that a system with matrix p shall exhibit that value upon measurement is... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

The trace of the matrix 2", i.e. 2 multiplied by itself n times, gives a sum of products, each composed of n [Pg.52]

PCDD/F and other chlorinated hydrocarbons observed as micropollutants in incineration plants are products of incomplete combustion like other products such as carbon monoxide, polycyclic aromatic hydrocarbons (PAH), and soot. The thermodynamically stable oxidation products of any organic material formed by more than 99% are carbon dioxide, water, and HCl. Traces of PCDD/F are formed in the combustion of any organic material in the presence of small amounts of inorganic and organic chlorine present in the fuel municipal waste contains about 0.8% of chlorine. PCDD/F formation has been called the inherent property of fire. Many investigations have shown that PCDD/Fs are not formed in the hot zones of flames of incinerators at about 1000°C, but in the postcombustion zone in a temperature range between 300 and 400°C. Fly ash particles play an important role in that they act as catalysts for the heterogeneous formation of PCDD/Fs on the surface of this matrix. Two different theories have been deduced from laboratory experiments for the formation pathways of PCCD/F ... 

In Liouville space, both the density matrix and the 4 operator become vectors. The scalar product of these Liouville space vectors is the trace of their product as operators. Therefore, the NMR signal, as a function of a single time variable, t, is given by (10), in which the parentheses denote a Liouville space scalar product ... 

The matrix Rij,kl = Rik Rjl represents the effect of R on the orbital products in the same way Rjk represents the effect of R on the orbitals. One says that the orbital products also form a basis for a representation of the point group. The character (i.e., the trace) of the representation matrix Rij,kl appropriate to the orbital product basis is seen to equal the product of the characters of the matrix Rjk appropriate to the orbital basis %e2(R) = Xe(R)%e(R)i which is, of course, why the term "direct product" is used to describe this relationship. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

If analysis is to be attempted with a detection system of only moderate selectivity, a substantial cleanup procedure may be required in order to enhance the concentration of the extracted trace residue while decreasing die concentration of possible interfering substances in the sample matrix. This is die case with most of the relatively nonspecific physicochemical detection systems used in residue analysis. Occasionally a sample may be suitable for direct physicochemical analysis after an extraction and concentration step. However, the majority of edible animal products need extensive cleanup to separate the compounds of interest from animal lipids and other natural organic substances prior to detection. For such detection systems, there has been a general rule dictating diat the cleaner sample, the better the result obtained. 

Finally, using the eigenvalues there are some further subdivision possible If the product of eigenvalues of a unitary matrix or operator is equal to +1, it is called a special unitary (SU) matrix or operator. Similar for real orthogonal matrices, where the only possible choice is +1 or -1 the former case is called special orthogonal (SO) matrices. For a matrix, this product equals the determinant of the matrix. For both matrices and operators, the sum of eigenvalues is called the trace of the matrix or operator. This equals the sum of the diagonal elements of a matrix representation. 

At time point f, the signal, fid(f), generated by a spin set can be determined by the trace of the product of the density matrix and the matrix of the I+ excitation operator ... 

Thus, the ensemble average of any observable quantity A can be obtained simply by taking the trace of the matrix product of A and p in either order. 

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