Lower triangle

After obtaining the described decomposition the set of equations can be readily solved. This is because all of the information required for transfonnation of the coefficient matrix to an upper triangular fonn is essentially recorded in the lower triangle. Therefore modification of the right-hand side is quite straightforward and can be achieved using the lower triangular matrix as... 

If Caf ) = C is a lower triangle matrix and all the operators Caa are invertible, then the procedure of solving equations (42 ) can be reduced to successive solution of the equations... 

In light of the special structure of the diagonal matrices k, whose blocks are lower triangle, the components s = 1,2,..., z, of the unknown vector yjQ.) are to be determined successively by the elimination method in passing from a to qH- 1 and from s to s- 1. By the elimination formulas for a three-point equation we constitute in a term-by-term fashion the vectors a = 1,2,..., p. Moving in reverse order from a -f 1 to a and from s -b 1 to s the vectors y(p+i), >y(2p) recovered from the system... 

As far as a non-self-adjoint operator B is concerned, the workable procedure g reduces to inversion of a lower triangle matrix. ... 

Fig. 3. Projection of the upper and the lower triangle of a geometry between an octahedron and a trigonal prism. Definition of the twist angle y>.

FIGURE 12. Spectra of vacidin A methoxycarbonylmethylamide 64. Spectral region 0.72-5.06 ppm of 300 MHz ROESY (upper triangle) and DQF-COSY (lower triangle) spectra of VacGlyOMe (15 mg ml-1, pyridine-d5-methanol-d4, 9 1) combined along the diagonal. Reproduced by permission of Japan Antibiotics Research Association from Reference 40... 

K D <-dist (X, method "binary") lower triangle matrix... 

Table 3 Overlap (upper triangle) and spin correlation (lower triangle) matrices successive values in the boxes relate to S = 0, 1, 2, and 3...

The differences ofthe actual values of (f, -sj) (lower triangle of Table 3) from the pure-singlet value of - also give the proportions of triplet character. The values highlighted... 

This system has the lower triangle Jacobian matrix dcii/daj. 

The Jacobian matrix (81) should have the lower triangle form in coordinates a, (75) for all nonnegative values of rate constants and concentrations. This is equivalent to the lower triangle form of all matrices M,y in these coordinates. Because usually there are many zero matrices among Mrj, it is convenient to describe the set of nonzero matrices. 

Special orthogonal matrices such as Householder matrices H = Im — 2vv for a unit column vector v G Cm with v = v v = 1 can be used repeatedly to zero out the lower triangle of a matrix Amn much like the row reduction process that finds a REF of A in subsection (B). The result of this elimination process is the QR factorization of Am,n as A = QR for an upper triangular matrix Rm,n and a unitary matrix Qm,m that is the product of n — 1 Householder elimination matrices Hi. 

Tab. 5-3. Correlation matrix for seven features (lower triangle)...

Table 6 Statistics on commutator HS — SH matrix elements for naphthalene. Lower triangle only. All are energies in Hartrees.

The VB calculations describing the 2B2 state were performed using at most seven atomic orbitals for the pair of structures. To form structures Ei and E2, we need two different orbitals on the central atom (atom 2 in Fig.5). These orbitals are shown in Fig. 12. One of them points toward the top atom and the other is spread inside the lower triangle. In this case, the orbitals are distinct, one being more delocalized than the other. The most diffuse orbital could be a typical "metallic orbital", because atom 2 is performing a metallic bond with atoms 3 and 4 through it. Only one orbital on the first atom ( atom 1 in Fig.5) is enough to form each pair of structures of the planar Li4 shown in Fig. 1 lb. On the other... 

Fig. 12. Two different orbitals on the central atom of the planar Li

It is a square unsymmetric Ax A matrix, where the upper triangle of the matrix contains the elements of the detour matrix (information about the longest paths) and the lower triangle contains the elements of the topological distance matrix (information about the shortest paths). 

Since this matrix is a sparse matrix, it is not economical to be explicitly used in computer programming. In order to save computer memory, the authors used the technique of the bond list, which encodes the position and the value of the non-zero elements in the upper-triangle (or lower-triangle) of the matrix. Thus, each member in the bond list is coded by (i,j, with i and j referring... 

When fhe lasf fwo indices are expanded from lower triangles to squares, symmetric and antisymmetric matrices are formed. The superscripts S and A denote these cases. Diagonal elements for the symmetric case have a different normalization factor. The rank of the spin-adapted H matrix for the closed-shell case is o- -v- -vo - -ov. ... 

Symmetric matrices have the property that XXT = I or XT = X. For simplicity, sometimes elements in the lower triangle are not shown, thereby implying the matrix is symmetric, e.g.,... 

---