As we have commented above, the triangular factors of the LDU-decomposition appearing when we apply complete pivoting to an n x n nonsingular matrix have off-diagonal elements with absolute value bounded above by 1 and then their condition numbers are bounded in terms of n.
Let us start by showing in the next result a first example, where pivoting is not necessary.
Gaussian elimination with a given pivoting strategy, for nonsingular matrices A = [([alpha].
t)] according to the given pivoting strategy and satisfying [[?
pivoting as a symmetric pivoting which chooses as pivot at the tth step (1 [less than or equal to] t [less than or equal to] n - 1) a row [i.
Although Gauss elimination without pivoting of a positive definite symmetric matrix is stable, it does not guarantee the well conditioning of the triangular factors.
The following example shows that applying symmetric complete pivoting to a Stieljes matrix does not guarantee that the triangular factors are diagonally dominant.
then the permutation matrix associated to the symmetric complete pivoting is the identity and A = [LDL.
pivoting interchanges the first and second rows and columns in the first step of Gauss elimination and it still produces another row and column exchange in the second step.
pivoting also can be applied to an n x n M-matrix A of rank r < n diagonally dominant by columns in order to obtain an upper triangular matrix [A.