Columns, deleting

Here, our idmissibility conditions read (8.1.6) and (8.1.12). Then, clearly, rankB = 3 is full row rank, and we have no redundancy. Now for observability. Deleting any of the columns m, m2, or mj leaves the rank unaltered. Let us delete the colunui T. We then obtain the matrix (8.5.3). If 0 0. on f3fwe have (8.1.18), hence the condition (8.1.13) with P instead of P is fulfilled and as above, B with column deleted remains with rank 3. According to (7.1.28), all the uiuneasured variables will be qualified as unobservable, whatever be an estimated value of m2 (> 0), and of P such that the inequality (8.1.13) is obeyed most likely, we shall use such estimates for a classification based on the linearized system. Let now, however, Q = 0. The matrix (8.5.7) is independent of this value and our first estimate of and P will probably lead to the same... 

Adjugate Matrix of a Matrix Let Ay denote the cofactor of the element Oy in the determinant of the matrix A. The matrix B where B = (Ay) is called the adjugate matrix of A written adj A = B. The elements by are calculated by taking the matrix A, deleting the ith row and Jth. column, and calculating the determinant of the remaining matrix times (—1) Then A" = adj A/lAl. This definition may be used to calculate A"h However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under Numerical Analysis and Approximate Methods. ... 

Modify Table) Once a data table is established, it can be manipulated in various ways The option offers Add/Delete Row/Column, and Change Entry choose the option and click on the appropriate item. If many rows or columns need to be added or deleted, it is easier to read the data file into Excel, do the modifications there, and reimport it using the (Import from Excel) option. 

After the chosen data file is read into MULTI, sequential columns K through L are selected for analysis K and L must be within the bounds 1. .. M, and L must be larger than K. If the columns one wants to analyze are separated by other data, use program DATA to first copy the unwanted column to column M + 1 by invoking the option (Add) and selecting the constant to be zero and then deleting the unwanted column. 

By contrast, Fig. 9c shows an alternative scheme using linked list. In this scheme (scheme II) the information associated with a nonzero element is stored in a triplet containing the row index, the value of the nonzero element, and a pointer to the address of the next element in the same column. The starting addresses of each column are stored in another n locations. Notice that in this scheme the successive elements need not be stored in consecutive locations. To insert or delete an element requires only the change of one or two pointers no rearrangement of the list is necessary. On the other hand, the storage requirement for the same matrix is now 3 N + n and, as it stands, to find a specific nonzero element requires a linear search through the chain. 

The sub-determinants, with appropriate signs, as they appear in the first stage of expansion of the determinant, are examples of cofactors. The minor, Mij of an element Oy is obtained by deleting the ith row and the jth column from the determinant, whereby the cofactor... 

Another way to calculate the value of a determinant is to evaluate its cofactors. The cofactor of an element a- of the matrix is found by first deleting from the original matrix the ith row and yth column corresponding to that element the resulting array is the minor (M. ) for that element and has dimension (n - 1) X (n - 1). The cofactor is defined as... 

To form the adjacency matrix in a simpler way, note that the elements of the first column of the adjacency matrix in Fig. 7b correspond exactly to the elements of the first column of the occurrence matrix in Fig. 6 if the output element is deleted. Similarly, the elements of column 2 of the adjacency matrix correspond exactly to the elements of the fourth column of the occurrence matrix if the output element is deleted. Therefore, if the columns of the occurrence were permuted until all of the output elements appeared on the main diagonal, as in Fig. 7a, the nonzero elements of the occurrence matrix... 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

The determinant, or distribution term for [E], for example, can be calculated from the coefficients listed above, after deleting the E column. For a mechanism of n intermediates, only n — 1 equations are needed. Thus, by leaving out the d[EP]/d row, we can write... 

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