In the context of inverse $N_0$-matrices, this study focuses on the closure of generalized Perron complements by utilizing the characteristics of $M$-matrices, nonnegative matrices, and inverse $N_0$-matrices. In particular, we illustrate that the inverse $N_0$-matrix and its Perron complement matrix possess the same spectral radius. Furthermore, we present certain general inequalities concerning generalized Perron complements, Perron complements, and submatrices of inverse $N_0$-matrices. Finally, we provide specific examples to verify our findings.

It is well known that there is an intrinsic connection between Perron complements and Schur complements. It has been demonstrated that Schur complements of strictly generalized doubly diagonally dominant matrices retain the property of strict generalized double diagonal dominance. Our primary aim of this study is to extend these findings to generalized Perron complements of nonnegative irreducible matrices. Specifically, we established that generalized Perron complements derived from strictly generalized doubly diagonally dominant and nonnegative irreducible matrices preserve strict generalized double diagonal dominance and nonnegative irreducibility. Numerical examples are provided to substantiate our theoretical results.