Linear algebra Gauss-Jordan elimination

This chapter gives a brief summary of properties of linear algebraic equation systems, in elementary and partitioned form, and of certain elimination methods for their solution. Gauss-Jordan elimination, Gaussian elimination, LU factorization, and their use on partitioned arrays are described. Some software for computational linear algebra is pointed out, and references for further reading are given. 

One method for finding A is called Gauss-Jordan elimination, which is a method of solving simultaneous linear algebraic equations. It consists of a set of operations to be applied to Eq. (9.51). In order maintain a valid equation, these operations must be applied to both sides of the equation. The first operation is applied to the matrix A and to the matrix E on the right-hand side of the equation, but not to the unknown matrix A . This is analogous to the fact that if you have an equation ax = c, you would multiply a and c by some factor, but not multiply... 

I.B.I. Compare the number of FLOPS necessary to solve a system of N linear algebraic equations by Gaussian and Gauss-Jordan elimination. Which one requires less work ... 

In general there are two ways to solve Eq. (L.3) for Xi,. . ., x elimination techniques and iterative techniques. Both are easily executed by computer programs. In the pocket in the back cover of this book you will find a disk containing Fortran computer programs that can be used in solving sets of linear equations. We shall illustrate the Gauss-Jordan eliinination method. Other techniques can be found in texts on matrices, linear algebra, and numerical analysis. 

In balancing proper, the (integral) reaction rates are certain parameters that can be eliminated see Section 4.3. The procedure is purely algebraic, based on the assumed stoichiometric matrix S of elements, thus S(n) in reaction node n. By Gauss-Jordan column elimination (4.3.9), rearranging possibly the rows of S we find matrix Sq (4.3.10) of Rq linearly independent columns... 

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