Elements, of matrix

Accordingly, let us assume that the entries in the matrices A and C are either zeros or arbitrary nonzero parameters. For nonlinear systems, their description will be accurate in an infinitesimal region around the point of linearization. Many of the elements of matrices A and C will vary from one linearization point to another, and some elements will always be zero. 

Given tv ro equal-sized graph-theoretical matrices Mi and M2, sum matrices, denoted as are obtained by summing corresponding elements of matrices Mi and M2 ... 

The elements of matrices W (z = A,. .., E) describe the fundamental curvature details. Hence, after enforcing the boundary constraints, one receives... 

In practice, we shall assume the existence of a set of matrix manipulation routines (which will he coded later) and only rarely have to access the individual elements of matrices. 

Figure 5.4 Partially structured tridiagonal block Jacobian. The points in row 941 and 942 and in column 941 and 942 represent the unstructured elements Specifically, the circles are elements of matrices A, 2 and A2,i, while the diamonds belong to the matrix A2,2.

The parameter n can be chosen equal to the common constant diagonal element of matrices f and f, since the resulting secular equation (11.60) will show pairing of roots around /i and the number of roots equals twice the number of electrons (cf. Eq. (11.70)). The pairing property can be shown from Eq. (11.61) by using the transformations... 

Here, matrix [AT] is a three by three (or 3X3) matrix whose elements are numbers, and L] is a three by one matrix with its elements representing variables x, y, and ss. The [AH is called a square matrix. A sqmre matrix has the same niunber of rows and columns. The element of a matrix is denoted by its location. For example, the element in the first row and the third column of matrix [AT] is denoted 1 i3, which has a value of nine. In this book, we denote the matrix by a boldfiice letter in brackets [] and, for example [AI], [T], F, and the elements of matrices are represented by regular lower case letters. The are used to distinguish a column matrix. A column matrix is defined as a matrix that has one column but could have many rows. On the other hand, a row matrix is a matrix that has one row but could have many columns. 

The interest of a Differential Formal Graph is to translate in a particularly simple way the notion of state function and to demonstrate in an elegant way the necessary equality between crossed coefficients, here mutual elastances, or in other words, nondiagonal elements of matrices. 

In this book several other sets of quantities occur which can conveniently be treated in matrix language. Thus the table of force constants fij for a given molecjule, in terms of any coordinate system, can be written in tabular form and the whole table called F, the force constant matrix. Similarly, the coefficients of various linear transformations and the quantum-mechanical integrals in Chap. 3 can be treated as elements of matrices. 

Quotient matrices have been introduced by Randic (1994b) and applied to structure-property modeling by Plavsid and his coworkers (Plavsic et al., 1998) and Nikolid and her coworkers (Nikolid et al., 2001a). The quotient matrices, denoted by M /M, are obtained by dividing the off-diagonal elements of matrices and M ... 

Following the PPP method in the case of molecules, we introduce systematically the neglect of differential overlap and, taking into account only first-neighbor interactions, the elements of matrices F( ) in expression (3.2) take the forms ... 

In the case of the CNDO/2 CO scheme for the valence electrons of one-dimensional periodic systems we can write equation (3.6), with expression (3.12) for H(it), if we apply again the first-neighbor-interaction approximation. In this case the expressions for the elements of matrices H( ) that can be derived again from the corresponding ab initio equations by introducing the simplifications and approximations of the CNDO/2 method, will be ... 

If the simplifications and parametrizations characteristic of the CNDO/2 method are again introduced, then the elements of matrices H( i, 2) in expression (3.18) reduce to... 

In order to calculate the elements of matrices P(4 ), a numerical integration over k in the interval (0, nta) is required. This was performed using Simpson s rule. 

The overlap matrix elements appearing in these expressions, combined with the elements of matrices in the terms like (39) give elements of matrices A, occurring in the definitions (36). 

These quantities are the diagonal elements of matrices of Pw, or pj, (ne-glecting a factor or —and can be interpreted as the average value of the angular momentum of the electronic motion corresponding to the Euler angles, or the average values of their squares. Then, in detail, Eq. (90) reads... 

Finding the elements of matrices B and G is a purely geometrical problem. The force constant matrix F is now a 3 x 3 matrix over the stretching and bending internal coordinates. A little algebra shows that the corresponding secular equation is now... 

Sensitivity matrices Sm = 9lnT,(t )/9lnjCi belong to a series of / time points within time interval [tj, tal, and the rows of these matrices belong to the variables present in the summation of the objective function in Eq. (5.15). From the calculation of matrix S S, the sum of the elements of matrices S , is obtained, and therefore the integral present in Eq. (5.15) is replaced by a summation in Eq. (5.16). 

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