Lower and upper triangular matrices

This technique (also known as the Grout reduction or Cholesky factorization) is based on the transfonnation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as... 

Therefore a = /nxl = hi, fl2i = hi, etc. (elements in the first column of a a,re the same as the elements in the first column of /) similarly multiplying rows of / by columns of u and equating the result with the corresponding element of a all of the elements of lower and upper triangular matrices are found. The general formula for obtaining elements of / and u can be expressed as... 

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. 

A given matrix A can be represented as a product of two conformable matrices B and C. This representation is not unique, as there are infinite combinations of B and C which can yield the same matrix A. Of particular usefulness is the decomposition of a square matrix A into lower and upper triangular matrices, shown as follows. 

We now extract from this matrix the lower and upper triangular matrices... 

We have seen that to make Gaussian elimination robust, we must include partial pivoting so that all kkj are finite. When the factorization is performed using Gaussian elimination with partial pivoting, the book-keeping is a bit more complex, but the result is similar We obtain lower and upper triangular matrices L and U, and a permutation matrix P, such that... 

Just as a matrix may be factored into ffie product of lower and upper triangular matrices, it may also be factored into the product of an orthogonal matrix Q, — Q, and an upper triangular matrix R,... 

However, as discussed in any text on linear algebra, it is inefficient to actually compute the matrix inverse. Rather, the A matrix is decomposed into the product of upper and lower triangular matrices,... 

Special orthogonal matrices such as Householder matrices H = Im — 2vv for a unit column vector v G Cm with v = v v = 1 can be used repeatedly to zero out the lower triangle of a matrix Amn much like the row reduction process that finds a REF of A in subsection (B). The result of this elimination process is the QR factorization of Am,n as A = QR for an upper triangular matrix Rm,n and a unitary matrix Qm,m that is the product of n — 1 Householder elimination matrices Hi. 

The LU decomposition to solve Ax = B given A and B, is developed in two steps. The two matrices L and U are determined from the equation LU = A and the requirement that U be upper triangular and L be lower triangular with unit diagonal elements. Thus for A an n x n matrix. A, L, and U have the form... 

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... 

Rather than directly calculating the product A b, we use a process called LU factorisation, to factor A as the product of two other matrices a lower and an upper triangular matrix, so that A = LU ... 

L,U] = LU(X) stores a upper triangular matrix in U and a "psychologically lower triangular matrix", i.e. a product of lower triangular eind permutation matrices, in L, so that X = L U. 

Tie + n = rip. In that case the nominal interaction forces can be uniquely determined. This can be done by applying a standard technique like first performing a decomposition of T into upper and lower triangular matrices T = LU and then solving the system by a forward and backward elimination. 

Outputs of the MATLAB function lu(A) are an upper triangular matrix V and a psychologically lower triangular matrix , that is, a product of lower triangular and permutation matrices, in L so that LV = A. 

For any upper triangular matrix, the eigenvalues are found on the principal diagonal. This also is true for lower triangular and diagonal matrices,... 

One of the classical methods for solving a set of linear equations such as Eq. (4.40) is to first convert the A matrix into the product of two special matrices LU where the L matrix is a lower triangular matrix and U is an upper triangular matrix with the forms ... 

A matrix of this form, that is, an upper and lower triangular quadrant for which no value is required (observed by the gray shaded area) is also known as a tri-diagonal matrix. More advanced methods of solving matrices (and in particular tri-diagonal types) are described in Burden and Faires (1997). ... 

Rg. 13.2. The lower triangular structure of the matrices T in (13.2.25). The diagonal blocks contain identity matrices, the lower triangular blocks are nonzero and the upper tiiangular blocks zero. 

For matrices that are not triangular, we cannot determine the eigenvalues by inspection, but we can obtain upper and lower bounds using Gershgorin s theorem. Let be an iVx W... 

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