Gauss—Jordan method

If det C 0, C exists and can be found by matrix inversion (a modification of the Gauss-Jordan method), by writing C and 1 (the identity matrix) and then performing the same operations on each to transform C into I and, therefore, I into C". ... 

Since det(A) =0 if and only if A is singular, it provides a convenient way of checking singularity. Determinants have traditionally been used also for solving matrix equations (ref. 10), but both the Gauss-Jordan method and the Gaussian elimination are much more efficient. The determinant itself can easily be calculated by LU decomposition. For the decomposed matrix (1.45)... 

A special property of solving a matrix equation in this way is that the LU decomposition does not involve the right-hand side vector b, in contrast both to the Gauss-Jordan method and to the Gaussian elimination. This is... 

Any standard method of matrix inversion, such as the Gauss-Jordan method (N13), may be used to solve the equations. The coefficients in equations 4.11-4.14 may be used without serious error for most ordinary Portland cement clinkers in which the alite composition is not too different from that assumed here. As a byproduct of the calculation described in this section, and using the full compositions of the phases given in Table 1.2, one may calculate a mass balance table (Table 4.3) showing the distributions of all the oxide components among the phases. 

If possible, bring 6i into the basis set 6 as the pivotal variable for row i and column i, by applying the modified Gauss-Jordan method described at the end of Section E.3 to the full A-matrix, including the last row and column. The requirements for such a move are LBAS(i) must be zero An must be positive (to ensure a descent of 5) Di and all resulting Dj must exceed ADTOL and every parameter must remain within the permitted region ofEq. (6.4-2). 

For computation of a particular solution vector x, this method requires i(n -n)-l-n operations of multiplication or division, versus n —n) +n for the Gauss-Jordan method. Thus, the Gauss method takes about two-thirds as many operations. The computation of an inverse matrix takes 0(n ) operations of multiplication or division for either method. 

If you use the Gauss-Jordan method outlined in Sec. L. 1, you will find that the transformed augmented matrix for the set of equations above in the format of Eq. (L.4) will have only zeros in one row, indicating that one equation is not independent and hence redundant. (As explained in Appendix L, if the determinant of the augmented matrix were not equal to zero, the rank of the augmented matrix... 

The essence of the Gauss-Jordan method is to transform Eq. (L.3) into Eq. (L.4) by sequential nonunique elementary operations on Eq. (L.3) ... 

To illustrate the elementary operations that are required to execute the Gauss-Jordan method, consider the following independent set of three equations in three unknowns ... 

A matrix is a list of quantities, arranged in rows and columns. Matrix algebra is similar to operator algebra in that multiplication of two matrices is not necessarily commutative. The inverse of a matrix is similar to the inverse of an operator. If A is the inverse of A, then A A = AA = E, where E is the identity matrix. We presented the Gauss-Jordan method for obtaining the inverse of a nonsingular square matrix. 

This is a systematic procedure for carrying out the method of elimination. It is very similar to the Gauss-Jordan method for finding the inverse of a matrix, described in Chapter 9. If the set of equations is written in the vector form... 

The primary use of the Gauss-Jordan method is to obtain an inverse of a matrix. This is done by augmenting the matrix A with an identity matrix I. After the elimination process in converting the matrix A to an identity matrix, the... 

Obtaining the matrix inverse using the Gauss-Jordan method provides a compact way of solving linear equations. For a given problem,... 

An advantage of this approach over the Gauss-Jordan method is that, in the Gauss-Jordan method, after a few samples have been selected the residual data represent the noise level and further selections are made essentially at random. In the approach using Mahalanobis distances, samples at the fringes of the data remain there no matter how many samples have already been... 

The overall system consists of Eqs. (2.30)- (2.36). This is a set of nonlinear simultaneous algebraic equations that can be solved simultaneously using a combination of Newton s method (Sec. 1.9) and the Gauss-Jordan method to be developed in this chapter (Sec. 2.6). 

U.se the Gauss-Jordan method to determine the values of the 14 unknown quantities jc,., 1 = 3.16. 

JORDAN Solves a set of linear algebraic equations by the % Gauss-Jordan method. 

A method closely related to Gauss elimination is called the Gauss-Jordan algorithm. As a bonus (but it involves more work), the inverse of the matrix is also calculated. The basic idea behind the Gauss-Jordan method is to hrst form an augmented matrix consisting of the original systan matrix and the identity matrix as follows ... 

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