A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semi-definite.

No (partial) pivoting is necessary for a strictly column diagonally dominant matrix when performing Gaussian elimination (LU factorization).

A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix) is non-singular.

The Jacobi preconditioner is one of the simplest forms of preconditioning, in which the preconditioner is chosen to be the diagonal of the matrix Assuming , we get It is efficient for diagonally dominant matrices .