Definition:Diagonally Dominant Matrix

Definition

Let $\mathbf A$ be a square matrix.

By Rows

$\mathbf A$ is diagonally dominant by rows if and only if:

in each row of $\mathbf A$, the absolute value of the element on the main diagonal is greater than the sum of the absolute values of the other elements on that row.

By Columns

$\mathbf A$ is diagonally dominant by columns if and only if:

in each column of $\mathbf A$, the absolute value of the element on the main diagonal is greater than the sum of the absolute values of the other elements in that column.

That is, if its transpose is diagonally dominant by rows.

Examples

Arbitrary Example

Consider the square matrix $\mathbf A$:

$\mathbf A = \begin {pmatrix} 2 & 1 & 0 \\ 1 & -3 & 1 \\ -2 & 1 & 4 \end {pmatrix}$

$\mathbf A$ is diagonally dominant by rows, but not diagonally dominant by columns.

Also see

• Results about diagonally dominant matrices can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): diagonally dominant matrix