Elementary row operations

The objective is to apply a sequence of elementary row operations (39) to equation 25 to bring it to the form of equation 22. Since the rank of D is 3, the order of the matrix is (n — r) x n = 2 x 5. The following sequence of elementary row operations will result in the desired form ... 

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 ... 

In this case, A can be transformed by elementary row operations (multiply the second row by 1/2 and subtract the first row from the result) to the unit-matrix or reduced I0W-echelon form ... 

These operations are called elementary row operations. For example, the linear system of Equations (7.9)... 

A particular sequence of elementary row operations finds special application in linear programming. This sequence is called a pivot operation, defined as follows. 

Fig. I. This diagonalized matrix (/ ,) is formed from matrix (ay) by elementary row operations and column permutations. It has the same rank as (cty).

Remark 7. The number of independent reactions in a given set lr is the rank of the stoicheiometric matrix and may be determined by elementary row operations. 

The problem may be restated now in geometrical language. The vector on the left side of Eq. (353) has six elements it will represent a vector in six dimensional space if none of the elements can be expressed as linear combinations of the other elements. On the other hand, if scheme (352) is to be equivalent to scheme (350), it must be a vector in five dimensional space. Hence, to prove the equivalence of schemes (350) and (352), we need only to show that the vector in Eq. (353) is really in five dimensions rather than six. This may be accomplished by showing that the 6X3 matrix in Eq. (353) can be transformed, by the elementary row operations (16) given below, into a matrix in which the third column is of the form... 

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. 

This is a system of three equations in three unknowns (Xj, X2, and X3), which can be solved by standard methods. Let us perform row operations on the above system to make the solution easier to interpret. Hence, performing elementary row operations and reducing the system to only elements on the main diagonal (also known as reduced row echelon form) gives... 

We emphasize again that the above system is an equivalent matrix that is as a result after elementary row operations have been applied to A (the right hand side remains unchanged because it is the zero vector, and it is unaffected by elementary row operations). This system may also be written out as a system of equations ... 

Inherently, the above system of equations suggests that there is only one degree of freedom (we can only specify one unknown before values for the other two unknowns can be calculated). Performing elementary row operations again we obtain... 

Note that Equation 6.7b is equivalent to Equation 6.7a after elementary row operations have been performed on it. Hence, now we can see the effect if we let Xj = 1... 

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... 

The rank of E is computed by standard methods, such as counting the number of independent rows after performing elementary row operations. We prefer here to employ MATLAB. Inputting E and computing its rank gives an answer of three. Since n = 3, rank(E) = n. The controllability matrix E has full row rank and the system is therefore controllable. 

A is a 3 X 5 matrix and thus the null space of A will be a two-dimensional subspace in c -Cb-Cc-Cd-Ce space (the size of matrix N must he nx(n- d), or 5x2). To compute the null space of this matrix, we can reduce A to reduced row echelon form by performing elementary row operations on A, and determine all of the vectors in the null space (similar in procedure to that shown in Example 3). Hence reducing A to the equivalent matrix gives ... 

This system contains a total of six unknowns in six equations. A beneficial property of this system, however, is that the volumetric flow rates and CSTR volumes appear linearly in the mass balance expressions. We therefore have a system of linear equations that must be solved. This may be done by performing elementary row operations on the appropriate matrices. 

Assume that r [Pg.95]

Form the 3x6 matrix [A /3] and transform it by elementary row operations to the form [/3 A]. The pivot element at each stage is emboldened. 

Gaussian elimination is considered the workhorse of computational science for the solution of a system of linear equations. Karl Friedrich Gauss, a great nineteenth-century mathematician, suggested this elimination method as a part of his proof of a particular theorem. Gaussian elimination is a systematic application of elementary row operations to a system of linear equations in order to convert the system to upper triangular form. Once the coefficient matrix is in upper triangular form. 

The elementary row operations are performed to put the augmented matrix into the upper triangular form (i.e., all elements below the diagonal are zero) ... 

To prove this formula, observe first that the system of equations for the stoichiometric coefficients always has the following particular solutions Sj = sj / 0, all other Si = 0 Sk = Sk 9 0, all other S = 0 and similarly for each catalyst. These solutions express the conservation of each catalyst and are independent, so that each catalyst accounts for at least one conservation conditions in fact, each catalyst accounts for one and only one conservation condition. For in the full matrix M derived from the equations for the stoichiometric coefficients, each catalyst determines a row whose only nonzero entries are 1 and —1, and since the contribution of this catalyst to any other conservation condition is always a multiple of this row, such contributions can always be removed by elementary row operations. Thus the system of equations for the stoichiometric coefficients can be reduced to a system in which certain equations do not contain the coefficients of any catalysts, whereas each of the remaining (independent) equations involves one and only one catalyst. This proves Eq. (2.18). 

Use elementary row operations to transform the augmented matrix into an upper-triangular one. 

Next, A is transformed into an identity matrix nsing elementary row operations (indicated by the matrix T) resnlting in... 

By performing elementary row operations on the [A /] matrix until it is transformed into the identity matrix, the following form results ... 

This procedure is known as an elementary row operation. The particular one shown here we denote by c x j + k. 

We now build a systematic approach to solving Ax = b based upon a sequence of elementary row operations, known as Gaussian elimination. First, we start with the original augmented matrix... 

Properties VI-IX give us the fastest method to compute the determinant. Note that the general formula for det( ) is a sum ofiV nonzero terms, each requiring Nscalar multiplications, and is therefore very costly to evaluate. Since Gaussian elimination merely consists of a sequence of elementary row operations that by property VII do not change the determinant... 

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