The \(\gamma \)-diagonally dominant degree of Schur complements and its applications

In this paper, we obtain a new estimate for the (product) \(\gamma \) $\gamma$ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly \(\gamma \) $\gamma$ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674).