Matrix lower triangular

After obtaining the described decomposition the set of equations can be readily solved. This is because all of the information required for transfonnation of the coefficient matrix to an upper triangular fonn is essentially recorded in the lower triangle. Therefore modification of the right-hand side is quite straightforward and can be achieved using the lower triangular matrix as... 

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... 

A.8 A technique called LU decomposition can be used to solve sets of linear algebraic equations. L and U are lower and upper triangular matrices, respectively. A lower triangular matrix has zeros above the main diagonal an upper triangular matrix has zeros below the main diagonal. Any matrix A can be formed by the product of LU. 

Using the normalized overlap formula, Eq. (68), the derivative with respect to the nonzero terms of the lower triangular matrix Lk is... 

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

The lower triangular matrix L can be constructed from the multipliers used in the elimination steps if we adjust them according to the rows interchanged. Taking into account that for the row of the pivot the multiplier is necessarily 1.0 (i.e., this row remains unchanged), in the three steps of the Gaussian... 

It then follows that the Lanczos expansion coefficients form a lower triangular matrix q = q with zero elements above the main diagonal [45] ... 

Predict the size distribution of the product of the breakage of broken particles. This product does not exist by itself as a separate entity, of course, because the particles that become broken remain mixed with those which stay unbroken. Even so, the distribution can be calculated by postmultiplying the n x n lower triangular matrix of breakage functions B in Table 13.4 by the percentage of particles broken, the nxl matrix (S)(F), as calculated in step 1. This postmultiplication works out as follows ... 

Here r is a 3n x 1 vector of Cartesian coordinates for the n particles, Lk is an n x n lower triangular matrix of rank n and I3 is the 3x3 identity matrix, k would range from 1 to A where N is the number of basis functions. The Kronecker product with I3 is used to insure rotational invariance of the basis functions. Also, integrals involving the functions k are well defined only if the exponent matrix is positive definite symmetric this is assured by using the Cholesky factorization LkL k. The following simplifications will help keep the notation more compact ... 

In applications one never computes the matrix L, but works with the lower triangular matrix L defined by the row interchange list p k) and the... 

The beauty of the REPSWA approach is that it can easily be employed to remove barriers in many degrees of freedom simultaneously without loss of efficiency. However, the method does require the evaluation of a Jacobian, which is not in general easy. Consider that proteins are polymers. It is, therefore, natural to consider growing in the barriers in the tjj angles one at a time. In this way, a lower/upper triangular matrix is formed. Since the Jacobian of an upper/lower triangular matrix is the product of its diagonal elements, REPSWA can be easily implemented. 

LU decomposition For a sqnare matrix A of order n, given that the determinants of the matrices Ap (p = l,2,...,n - formed by the elements at the intersection of the first p rows and columns of A are nonzeroes, then there exists a unique lower triangular matrix L and a unique upper triangular matrix U such that... 

A lower triangular matrix is the opposite of an upper triangular matrix, e.g.,... 

Let L be the lower triangular matrix formed by use of the multipliers of part (a) and by the use of elements of unity along the central diagonal, that is,... 

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