---
tag: 89P5GTKW87
kind: concept
name: Diagonally dominant matrix
msc: [15B99, 65F05]
specializes: [0JXZKX526R]
sources:
  - url: https://en.wikipedia.org/wiki/Diagonally_dominant_matrix
same_as:
  wikidata: Q952222
  wikipedia: Diagonally dominant matrix
---

## Summary

A square matrix is diagonally dominant when each row has a large diagonal entry.
In absolute value, it is at least the sum of the absolute values of the other entries of its row.
It is strictly diagonally dominant when this inequality is strict in every row.
A strictly diagonally dominant matrix is invertible, by the [[Levy–Desplanques theorem|PWKN6ANA0G]].
For strictly diagonally dominant matrices, Gaussian elimination needs no pivoting, and the Jacobi and Gauss–Seidel iterations converge.

## Explanations

### L1 human

In a diagonally dominant matrix, each row is controlled by its diagonal entry: the other entries together cannot outweigh it.
For example, $\begin{pmatrix} 4 & 1 & -2 \\ 1 & 3 & 1 \\ 0 & 2 & 5 \end{pmatrix}$ is diagonally dominant, since $4 \ge 3$, $3 \ge 2$ and $5 \ge 2$.
Such a matrix behaves much like a diagonal matrix, which is easy to invert.
This makes it easy to see invertibility and to solve systems with it reliably.