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 greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row. More precisely, the matrix ${\displaystyle A}$ is diagonally dominant if

${\displaystyle |a_{ii}|\geq \sum _{j\neq i}|a_{ij}|\ \ \forall \ i}$

where ${\displaystyle a_{ij}}$ denotes the entry in the ${\displaystyle i}$th row and ${\displaystyle j}$th 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.^[1]