Diagonal Dominance

A square matrix $A$ is diagonally dominant if, for every row $i$, the magnitude of the diagonal entry is greater than or equal to the sum of the magnitudes of all off-diagonal entries in that row:

$$|a_{ii}|\ge \sum _{j\ne i}|a_{ij}|\text{for all }i$$

Significance

Diagonal dominance guarantees the convergence of iterative solvers such as Liebmann’s method (Gauss-Seidel iteration). The matrix arising from the Laplacian difference equation satisfies this property: the diagonal entry is $-4$ (or $4$ in magnitude), and the four off-diagonal entries each have magnitude 1, so $4\ge 4$ with equality - the matrix is weakly diagonally dominant.

For the BTCS scheme, the tridiagonal matrix has diagonal entries $1+2c$ and off-diagonal entries $-c$, giving $1+2c\ge 2c$, which holds strictly. This is one reason the BTCS system is well-conditioned and solvable.

Liebmann’s method | Laplacian difference equation | BTCS scheme