Matrix algebra conformability

In matrix algebra it is possible for the product of two conformable matrices, neither of which is a null matrix, to be a null matrix. For example, if A and B are... 

Flory reviewed in 1969 the development and applications of the rotational isomeric state scheme calculations, which allow, by matrix algebra, the statistical mechanical averaging over the rotational states of chain properties which may be expressed as a vector or tensor quantity associated with the chain bonds, and estimations of the probabilities of chosen conformational sequences. The methods were generalized and schemes for reducing the dimensions of certain generator matrices were presented in 1974, when comparisons were also made with an alternative Fourier expansion method, currently in use for atactic polypropylene. These techniques have greatly contributed to an understanding... 

For all the afore-mentioned computations of Kx values Semiyen et al. used Eq. (5.24). The unperturbed mean square end-to-end distance of the linear precursors were calculated via the matrix algebraic methods of Flory and Jemigan [84, 85] using rotational isomeric state models based on detailed structural information [86]. However, Semiyen et al. [62, 63] also improved and applied another mathematical approach to the calculation of Kx, the so-called Direct Computational Method . The JS theory is limited to polymers obeying Gaussian statistics and cycles, free of enthalpic interactions. The Direct Computational Method does not need such restrictions [87-90]. The distances between terminal atoms of chains are calculated for all discrete conformations defined by the rotational isomeric state model. Any correlation between the directions of terminal bonds involved in the cyclization process can be investigated and their effect on Kx assessed. It can take into account favorable and unfavorable correlations between the directions of terminal bonds, as well as any excluded volume effect. Semiyen demonstrated [62, 63, 72] that the Direct Computational Method yields more realistic Kx values for small cyclic oligomers. 

Next, if g is a subalgebra of the conformal algebra c(l,3) with a nonzero projection on the vector space spanned by the operators D. Kq. K. Kt. K, then the corresponding matrices are linear combinations of the matrices E and. Sj(v. That is why the matrix should be sought in the more general form... 

To further emulate ordinary scalar algebra, we require the operation of matrix multiplication. The product AB of conformable matrices A, B (i.e., with nr of B equal to nc of A, as happens automatically for square matrices of dimension /) is defined by... 

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