  In matrix theory and numerical analysis there are two very famous and
important results. One is Gersgorin circle theorem, the other is strictly
diagonally dominant theorem. They have important application and research
value, and have been widely used and studied. In this paper, we investigate
generalized diagonally dominant matrices and matrix eigenvalue inclusion
regions. A class of G-function pairs is proposed, which extends the concept of
G-functions. Thirteen kind of G-function pairs are established. Their
properties and characteristics are studied. By using these special G-function
pairs, we construct a large number of sufficient and necessary conditions for
strictly diagonally dominant matrices and matrix eigenvalue inclusion regions.
These conditions and regions are composed of different combinations of
G-function pairs, deleted absolute row sums and column sums of matrices. The
results extend, include and are better than some classical results.
