Elimination Gauss-Jordan

Gauss-Jordan elimination is a variation of the Gauss elimination scheme. Instead of obtaining the triangular matrix at the end of the elimination, the Gauss-Jordan has one extra step to reduce the matrix A to an identity matrix. In this way the augmented vector b is simply the solution vector x. 

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 

Interchange the first and the second row to make the pivot element having the largest magnitude hence, we have 

scale the pivot element to unity (this step is not in the Gauss elimination scheme) to give 

Following the same procedure of Gauss elimination with the extra step of normalizing the pivot element before each elimination, we finally obtain 

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  

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

If the original square matrix, A, is given by the following expression  

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

To solve the system of equations Ax = b, use the same set of operations, indicated as the transformation matrix T, as follows  

---

Gauss-Jordan elimination is a variation of the preceding method, which by continuation of the same procedures yields... 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

The classification procedure developed by Madron is based on the conversion, into the canonical form, of the matrix associated with the linear or linearized plant model equations. First a composed matrix, involving unmeasured and measured variables and a vector of constants, is formed. Then a Gauss-Jordan elimination, used for pivoting the columns belonging to the unmeasured quantities, is accomplished. In the next phase, the procedure applies the elimination to a resulting submatrix which contains measured variables. By rearranging the rows and columns of the macro-matrix,... 

In this section we restrict considerations to an nxn nansingular matrix A. As shown in Section 1.1, the Gauss-Jordan elimination translates A into the identity matrix I. Selecting off-diagonal pivots we interchange some rows of I, and obtain a permutation matrix P instead, with exactly one element 1 in each row and in each column, all the other entries beeing zero. Matrix P is called permutation matrix, since the operation PA will interchange some rows of A. ... 

Solution of matrix equations bv Gauss-Jordan elimination... 

Example 1.1.4 Inversion of a square matrix try Gauss-Jordan elimination. 

REH El. 1.1.4. INVERSION OF A NATRIX BY GAUSS-JORDAN ELIMINATION 104 REN HERGE N10... 

The use of the proposed iterative method reduced the error below 10" in all the cases, which means that a solution with at least 10 accurate digits has been obtained. The solution for x. obtained in this way has been compared to the solution calculated by different methods (i.e. Gauss-Jordan elimination). The results of the comparison verified the above conclusion. 

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. 

For relatively simple networks, observation allows selection of reactions that are independent—for more complex systems use the Gauss-Jordan elimination to reduce the network to a set of independent (nonzero rows) reactions. 

When the matrix is reduced to echelon form by Gauss-Jordan elimination, the rank of the matrix can be shown to be equal to 3. With n = 5, the number of independent reactions is 5 - 3 = 2. Equation (4.575) requires that, for each of the two independent reactions,... 

The inverse of a matrix obeys A A = AA = E where E is the identity matrix. The inverse of a given matrix can be obtained by the Gauss-Jordan elimination procedure. 

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... 

Row operations are carried out on this augmented matrix a row can be multiplied by a constant, and one row can be subtracted from or added to another row. These operations will not change the roots to the set of equations, since such operations are equivalent to multiplying one of the equations by a constant or to taking the sum or difference of two equations. In Gauss-Jordan elimination, our aim is to transform the left part of the augmented matrix into the identity matrix, which will transform the right column into the four roots, since the set of equations will then be... 

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... 

Solve the set of equations, using Gauss or Gauss-Jordan elimination. 

Typical procedures to solve the OLS problem are Gaussian elimination and Gauss-Jordan elimination. More efficient solutions are based on decomposition of the X matrix by algorithms, such as LU decomposition. Householder reduction, or singular value decomposition (SVD). One of the most powerful methods, SVD, is outlined as follows (cf. Section 5.2 and Biased Parameter Estimations PCR and PLS Section). 

A related method is Gauss-Jordan elimination, which proceeds the same way as Gaussian elimination, except that instead of eliminating X2 from equations 3,4,..., n. 

An efficient way to solve a system of linear homogeneous equations is to do Gauss-Jordan elimination on the equations. If only the trivial solution exists, the final set of equations obtained will be = 0, X2 = 0,... = 0. If a nontrivial solution exists,... 

To eliminate one set of divisions, we interchange the first and second equations so that we start with an = 1. Detadiing the coefficients and proceeding with Gauss-Jordan elimination, we have... 

---