Definition – Diagonally dominant matrix.

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1. Definition

Definition

Let $A$ be an $n\times n$ matrix, where $n$ is finite, let $A_i$ denote the set of all elements of row $i$ of $A$ , and let $a_{ij}$ denote the element in row $i$ and column $j$ of $A$ . Finally, let $B_i=A_i\setminus{\{a_{ii}\}}$ . We say that $A$ is diagonally dominant if, for every element $a_{ii}$ of the leading diagonal,

$|a_{ii}|\geq \sum\limits_{b\in{B_i}} |b|$

And, there exists at least one $a_{ii}$ for which

$|a_{ii}| > \sum\limits_{b\in{B_i}} |b|$