A diagonally dominant matrix is a square matrix 
such that 
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.
