Diagonally dominant matrix

A square matrix is diagonally dominant when the absolute value of every diagonal entry is at least the sum of the absolute values of the other entries in its row.

A matrix is strictly diagonally dominant when every diagonal-dominance inequality is strict. A real symmetric strictly diagonally dominant matrix with positive diagonal is positive definite by the Gershgorin circle theorem.

If $\delta ={\min }_i(|a_{ii}|-{\sum }_{j\ne i}|a_{ij}|)>0$, then $A$ is invertible and its operator norm on ${\ell }^{\infty }$ satisfies the displayed bound. For $Ax=b$, choose $i$ maximizing $|x_i|$. The reverse triangle inequality gives $|b_i|\ge \delta \Vert x{\Vert }_{{\ell }^{\infty }}$. Taking $b=0$ proves injectivity and hence invertibility in finite dimensions; taking arbitrary $b$ proves the bound. The estimate is useful for converting strict diagonal dominance into quantitative matrix inverse stability.