Linear and multilinear algebra; matrix theory Definition

Diagonally dominant matrix

Definition

Let $A = (a_{ij})$ be an $n \times n$ complex square matrix. The matrix $A$ is diagonally dominant (by rows) when $\abs{a_{ii}} \ge \sum_{j \ne i} \abs{a_{ij}} \qquad \text{for all } i = 1, \dots, n.$ It is strictly diagonally dominant when this inequality is strict for all $i$ .

Remarks

Example: $\begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$ is strictly diagonally dominant, because $3 > 1$ and $2 > 1$ . Non-example: $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ is not diagonally dominant, because in the first row $\abs{1} < \abs{2}$ . Diagonal dominance by columns is the same condition for the transpose $A^T$ . A strictly diagonally dominant matrix is invertible, by the Levy–Desplanques theorem. A diagonally dominant matrix that is not strictly so can be singular, for example $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ .

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Square matrix concept
Diagonally dominant matrix definition

Tag	`YR2MKGA2CE`
URI	`m2://YR2MKGA2CE`
Kind	definition
Writing	W0 stub
MSC	15B99, 65F05
Defines	Diagonally dominant matrix
Uses	Square matrix
Changed	2 November 2026

Diagonally dominant matrix

Definition

Remarks

Reading path

Further reading

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