Useful Row Operations

The inverse in matrix algebra plays a similar role to division in scalar division. The inverse is defined as follows  

But since the multiplication is associative, the above equation will become 

The analytical technique according to the Gauss-Jordan procedure for obtaining the inverse will be dealt with later. 

A given matrix A can be represented as a product of two conformable matrices B and C. This representation is not unique, as there are infinite combinations of B and C which can yield the same matrix A. Of particular usefulness is the decomposition of a square matrix A into lower and upper triangular matrices, shown as follows. 

This is usually called the LU decomposition and is useful in solving a set of linear algebraic equations. 

---

It is assumed that the reader is somewhat familiar with this technique and only a brief review of the method is given here. In this approach, one uses row operations, sueh as multiplication by constants and additions of rows to transform the original matrix into a new matrix whieh has zeros for all the off-diagonal elements below the diagonal elements. This is illustrated below for a 5 by 5 set of equations ... 

We can use elementary row operations, also known as elementary matrix operations to obtain matrix [g p] from [A c]. By the way, if we can achieve [g p] from [A c] using these operations, the matrices are termed row equivalent denoted by X X2. To begin with an illustration of the use of elementary matrix operations let us use the following example. Our original A matrix above can be manipulated to yield zeros in rows II and III of column I by a series of row operations. The example below illustrates this ... 

As we have just shown using two series of row operations we have... 

Hopefully Chapters 1 and 2 have refreshed your memory of early studies in matrix algebra. In this chapter we have tried to review the basic steps used to solve a system of linear equations using elementary matrix algebra. In addition, basic row operations... 

These two matrices (original and final) are row equivalent because by using simple row operations the right matrix was formed from the left matrix. The final matrix is equivalent to a set of equations as shown below ... 

Table 5-2 Results after substituting into the original equations and calculating the differences between predicted and actual results (using manual row operations)...

Thus we see that we cannot arbitrarily select any subset of the data to use in our computations it is critical to keep all the data, in order to achieve the correct result, and that requires using the regression approach, as we discussed above. If we do that, then we find that the correct fitting equation is (again, this system of equations is simple enough to do for practice - the matrix inversion can be performed using the row operations as we described previously) ... 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

Row operations can also be used to obtain an inverse matrix. Suppose we augment A with an identity matrix I of the same dimension then multiply the augmented matrix by A-1 ... 

The determinant of A is unchanged by the row operations used in Gaussian elimination. Take the first three columns of C3 above. The determinant is simply the product of the diagonal terms. If none of the diagonal terms are zero when the matrix is reformulated as upper triangular, then A = 0 and a solution exists. If A = 0, there is no solution to the original set of equations. 

Using row and column operations, convert the following 7-factor Plackett-Burman design to the saturated fractional factorial design shown in Table 14.7. 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

The negative value of the multiplier 1/3 has been stored in the location where a zero was obtained by use of this multiplying factor. To emphasize that a zero and not —1/3 is to be used in subsequent row operations, the multiplier is enclosed by parentheses. 

The steps in the LU factorization of A by use of row operations as demonstrated in Chap. 4 follow together with the vector containing the pivot row information. First, the original matrix with its corresponding IPV vector is... 

The three transformed matrices Alf A2, and A3 are stored as linked lists in Fig. 15-2. To verify the fact that the linked list in Fig. 15-2 contains the elements of the transformed matrices Alt A2, and A3, an analysis analogous to that used to confirm the fact that the linked list in Fig. 15-2 contains the elements of A [Eq. (15-1)] may be performed. By comparison of Figs. 15-1 and 15-2, the fill resulting from the row operations is readily determined. The above formulation of a linked list storage procedure follows closely that used by Gallun.4... 

Use Gauss-Jordan elimination to solve the set of simultaneous equations in the previous exercise. The same row operations will be required that were used in Example 9.10. Q... 

The terms are present because an orthogonalized operator basis set is not employed in these calculations. The last row of Table III presents results for calculations that include both the third-order contributions to the partitioned EOM equation due to single excited configurations and use an operator basis in which the 3-block operators are Schmidt-orthogonalized to the 1-block (therefore, D - =0) through second order. These results differ from the results using the nonorthogonal operator basis by only 0.0 to 0.01 eV for each symmetry. 

The computations of the method are rooted in dual linear programming formulation and row operations similar to ones used in Simplex algorithm to solve linear programming problems. We can apply transportation algorithm using the initial solution firom Vogel approximation method. As mentioned in Taha s book [3] multipliers , and y, are associated with row i and column j of transportation table. 

Since there is only one column in A (corresponding to a single reaction), rank(A) = 1. To compute the set of concentrations orthogonal to the stoichiometric subspace, we compute the null space of A . Hence, since the rank of A is one, we expect the rank of the null space to be (3 - 1) = 2. We may compute the null space using standard methods such as elementary row operations. It... 

---