Diagonally dominant matrix

In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if

${\displaystyle |a_{ii}|\geq \sum _{j\neq i}|a_{ij}|\quad {\text{for all }}i\,}$

where a_ij denotes the entry in the ith row and jth column.

This definition uses a weak inequality, and is therefore sometimes called weak diagonal dominance. If a strict inequality (>) is used, this is called strict diagonal dominance. The unqualified term diagonal dominance can mean both strict and weak diagonal dominance, depending on the context.