4.7 Diagonal Dominance and Pivoting

We will begin our discussion of pivoting by identifying a condition in which pivoting is unnecessary. The matrix A is row diagonally dominant when the following inequality holds.

|aii|>ji|aij|,wherei=1,,n\left|{a}_{ii}\right|>\sum_{j\neq i}\left|{a}_{ij}\right|,\operatorname{where}% i=1,\cdots,n (36)

The matrix A is column diagonally dominant when the following inequality holds.

|ajj|>ji|aij|,wherej=1,,n\left|{a}_{jj}\right|>\sum_{j\neq i}\left|{a}_{ij}\right|,\operatorname{where}% j=1,\cdots,n (37)

If either of these conditions apply, the LU decomposition algorithms discussed in this document are numerically stable without pivoting.