Upper triangular form

Let us suppose that we can convert the n x coefficient matrix in equation system (6.5) into an upper triangular form as... 

Next, with the second row in C2 kept intact, the upper triangular form is achieved by operating on the third row ... 

The next pivot will be [A]4 3 = 4.0, thereby interchanging rows 3 an 4. To eliminate xj from equation 3 we need the single multiplier -2.0/4.0 = -0.5, and obtain the matrix in the desired upper triangular form ... 

Each set of values /ajg for normal mode p forms a N x N matrix, where N is the number of nuclei. A diagonal term Jaa p represents the contribution which atom a makes to Jp, and the sum Ja/3 p + the contribution due to the coupled motion of the pair of nuclei a and / . The graphical representation as full and empty circles, depending on the sign, in a matrix, in upper triangular form, maps the way nuclear motion creates Raman and ROA intensity in the vibrating molecule [42],... 

The model equations result in an upper-triangular form for the matrix A of Equation 18.6, which may be consequently exploited to construct the general solution. Starting with the equation for i = n and proceeding backwards, the general solution may be determined implicitly as... 

There is a similar procedure known as Gauss elimination, in which row operations are carried out until the left part of the augmented matrix is in upper triangular form. The bottom row of the augmented matrix then provides the root for one variable. This is substituted into the equation represented by the next-to-bottom row, and it is solved to give the root for the second variable. The two values are substituted into the next equation, and so on. 

The matrices Q and U are obtained in the procedure of reduction of the matrix A using a series of orthogonal transformations, like Householder reflections (Householder 1958 Wilkinson 1965 George and Ng 1986) or Givens rotations (Givens 1954 Wilkinson 1965, George and Heath 1980), on the upper triangular form. 

George A., Ng E. (1986) Orthogonal Reduction of Sparse Matrices to Upper Triangular Form Using Householder Transformations. SIAM J. Sci. Stat. Comput., Vol. 7, No. 2, pp 460-472. 

Both systems have been simultaneously reduced to upper triangular form, so back substitution immediately gives the solutions X2 = -3, xi = -2 to the first system and JC2 = —10, Xi = —7 to the second. This defines the inverse matrix A = (Verity that AA = A A = I... 

In fact, we need not compute the inverse B and can find the solution directly on rearranging matrix B into upper-triangular form. The procedure is again based on the knowledge of p n) and j n). By the one-to-one correspondence, we have... 

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) ... 

An important application of a basis change occurs when a given set of functions <, is not orthonormal for a new basis set <, that is orthonormal may evidently be constructed according to (2.2.16) in an infinite variety of ways. One common choice is to take T in upper-triangular form so that... 

Gaussian elimination to place Ax = b in upper triangular form... 

The following algorithm transforms a linear system to upper triangular form by Gaussian elimination. ... 

A matrix A is irreducible if there exists no permutation matrix P, such that PAP takes the block upper triangular form... 

As we demonstrate below for an example matrix, as oo, this sequence of matrices becomes of block upper triangular form,... 

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