Let \(A\) be an \(n\times n\) complex matrix and let \(0 \leq \alpha \leq 1\). If \[ | a_{ii} | \geq \Biggl( \sum^ n_{j=1\atop j \neq i}| a_{ij}|\Biggr)^ \alpha\Bigg(\sum^ n_{j=1\atop j\neq i}| a_{ji}|\Bigg)^{1-\alpha},\quad i = 1,\dots,n, \] then \(A\) is said to be \(\alpha\)-diagonally dominant, and when the strict inequalities hold \(A\) is called strictly \(\alpha\)-diagonally dominant. If there exists a positive diagonal matrix \(D\) such that \(AD\) is strictly \(\alpha\)-diagonally dominant, then \(A\) is a generalized strictly \(\alpha\text{-diagonally}\) dominant matrix.
The author presents a necessary and sufficient condition for an \(\alpha\)- diagonally dominant matrix to be a generalized strictly \(\alpha\)- diagonally dominant matrix. As applications of this result, sufficient conditions are given for a matrix to be a nonsingular \(M\)-matrix or a negatively stable matrix.