Matrix echelon

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

The echelon form of the equations can also be put into matrix form as follows. Echelon form ... 

The first (preceding) matrix is reduced to echelon form (zeros in the first and the second rows of column one) by... 

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 ... 

The system of equations (2.12) can be written using the column echelon form of matrix M as follows ... 

The rank of matrix M is 7. As the system is rank deficient, it admits a decomposition into two subsystems, one estimable and the other nonestimable. To determine which variables are observable, the column echelon form of M is obtained and T l... 

The utility of matrices in the applied sciences is, in many cases, connected with the fact that they provide a convenient method for the formulation of physical problems in terms of a set of equations. It is therefore important to become familiar with the manipulation of the equations, or equivalently with the manipulation of rows and columns of the corresponding matrix. First, we will be concerned with some basic tools such as column-echelon form and elementary matrices. Let us introduce some definitions (Noble, 1969). 

A matrix is said to be in column echelon normal form if ... 

Although there is a considerable degree of freedom in the sequence of calculations, when reducing a matrix to a column-echelon form, this is unique and the rank of the matrix is equal to the number of nonzero columns in the column echelon. 

If G is an (m x g) matrix of rank k and Ug denotes the column-echelon form of G, then a nonsingular matrix Eq exists such that... 

The column-echelon form of G is obtained by performing a sequence of elementary column operations on this matrix. This means that we can find a sequence of elementary matrices EpEp i... Ei corresponding to the elementary column operations, such that... 

Also, (2) follows immediately since the column-echelon form of a nonsingular matrix is the unit matrix. ... 

The row operations (a) to (c) are performed on (A b) until the front m by n matrix A achieves row echelon form. In a row echelon form R of A each row has a first nonzero entry, called a pivot, that is further to the right than the leading nonzero entry (pivot) of any previous row, or it is the zero row. 

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 rank of a matrix A is the number of nonzero rows in the reduced row echelon form of A. 

The echelon is now one row smaller than the echelon, Eq. (13.19), but the row with the zeros disappeared and the echelon is of the same form. Therefore, we have in the matrix one equation that can be deduced by a combination of the other equations. 

Of course, first we need to find an initial minimal DD pair. Following the null space approach [23, 24], we compute a basis of the kernel of the stoichiometric matrix S. More specifically, we compute a column-reduced echelon form of the basis and (after a permutation of rows) obtain... 

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 ... 

The matrix U has rank r, so there will be r basic variables and (n - r) free variables in the solution for h. In fact, we may further reduce the system in Equation 5.72 into row-reduced echelon form as follows ... 

An augmented molecular matrix can be transformed to a Reduced Row Echelon Form or RREF. This method is essential to all matrix transformations in this chapter. The idea behind the RREF is that we work from the first colunrn all the way to the rightmost one. For each column we determine whether it is possible to eliminate it by finding a nonzero entry, or pivot, in a row that has not been considered before. If not, we skip to the next column. If a pivot is found, we use it to eliminate all other entries in that row. We also move the pivot row up as far as possible. We cannot tell in advance where all the pivots will be found we must find them one by one since the elimination procedure can change zero entries into nonzero ones and vice versa. In general, we also do not know in advance how many pivots will be found. However, in the special case of a matrix augmented with a unit matrix, we do know that their number will be equal to the number of rows. 

By elimination, the number of independent equations can be determined. Only when the number of variables equals the number of equations, while the equations are independent, can a non-singnlar solntion be fonnd. In that case the eliminated matrix is a full square matrix. In this context the concept rank is nseful. The rank of a matrix is the maximum number of independent rows (or, the maximnm nnmber of independent columns). A square matrix A( ,n) is non-singular only if its rank is equal to n. The rank can easily be found from the number of non-zero rows obtained by redncing the matrix to echelon form. 

An efficient implementation of the matrix products requires that the sparsity pattern of the matrices (or more exactly, its echelon form) be considered. The matrices usually have wide portions of zeros Aat can be exploited to speed up the computations. Furthermore, the matrix R M R may have regions of zeros that can be considered by using a skyline profile solver. 

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