A square matrix is called diagonally dominant if for all . is called strictly diagonally dominant if for all .
A strictly diagonally dominant matrix is nonsingular. A symmetric diagonally dominant real matrix with nonnegative diagonal entries is positive semidefinite.
If a matrix is strictly diagonally dominant and all its diagonal elements are positive, then the real parts of its eigenvalues are positive; if all its diagonal elements are negative, then the real parts of its eigenvalues are negative. These results follow from the Gershgorin circle theorem.