The definition in the first paragraph sums entries across rows. It is therefore sometimes called row diagonal dominance. If one changes the definition to sum down columns, this is called column diagonal dominance.
The definition in the first paragraph uses a strict inequality. It is therefore sometimes called strict diagonal dominance. If a weak inequality () is used, this is called weak diagonal dominance.
If an irreducible matrix is weakly diagonally dominant, but in at least one row (or column) is strictly diagonally dominant, then the matrix is irreducibly diagonally dominant.
A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semi-definite. If the symmetry requirement is eliminated, such a matrix is not necessarily positive semi-definite; however, the real parts of its eigenvalues are non-negative.
Many matrices that arise in finite element methods are diagonally dominant.
A slight variation on the idea of diagonal dominance is used to prove that the pairing on diagrams without loops in the Temperley-Lieb algebra is nondegenerate. For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of appearing in each row appears only on the diagonal. (The evaluations of such a matrix at large values of are diagonally dominant in the above sense.)