The matrix is strictly diagonally dominant because The following results can be proved trivially from Gershgorin's circle theorem.
with real non-negative diagonal entries is positive semidefinite.
If the symmetry requirement is eliminated, such a matrix is not necessarily positive semidefinite.
For example, consider However, the real parts of its eigenvalues remain non-negative by Gershgorin's circle theorem.
No (partial) pivoting is necessary for a strictly column diagonally dominant matrix when performing Gaussian elimination (LU factorization).
The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant.
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 non-degenerate.
[3] For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of