Prerequisite for some easy Gaussian elimination A

a zero pivot is never encountered when applying Gaussian elimination with Type III operations,

LU factorization

Property of L

L has the remarkable property that below its diagonal, each entry 'ij is precisely the multiplier used in the elimination (3.10.1) to annihilate the (i, j)-position. This is characteristic of what happens in general.

elementary lower triangular matrix

Uniqueness of LU

The matrices L and U are uniquely determined by properties (3.10.7) and (3.10.8).

The decomposition of A into A = LU is called the LU factorization of A, and the matrices L and U are called the LU factors of A.

Computing A^−1

.

existence of LU factors.

leading principal submatrices of A

....Each leading principal submatrix Ak is nonsingular

Row interchanges LU decomposition

i.e for previous multiple lists, the multiples in each step is also interchanged according to the current interchange of rows.

Consequently, we may conclude that for every nonsingular matrix A, there exists a permutation matrix P (a product of elementary interchange matrices) such that PA has an LU factorization.

The LDU factorization

when A is symmetric,

the LDU factorization is

The Cholesky Factorization

positive definite.

.

A symmetric matrix A possessing an LU factorization in which each pivot is positive is said to be positive definite.

A is positive defifinite if and only if A can be uniquely factored as

where R is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky factorization of A, and R is called the Cholesky factor of A.