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

A

{\displaystyle A}

is diagonally dominant if

|

a

i

i

|

≥

∑

j

≠

i

|

a

i

j

|

∀

i

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

where

a

i

j

{\displaystyle a_{ij}}

denotes the entry in the

i

{\displaystyle i}

th row and

j

{\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.

Variations

The definition in the first paragraph sums entries across each row. It is therefore sometimes called row diagonal dominance. If one changes the definition to sum down each column, this is called column diagonal dominance.

Any strictly diagonally dominant matrix is trivially a weakly chained diagonally dominant matrix. Weakly chained diagonally dominant matrices are non-singular and include the family of irreducibly diagonally dominant matrices.