ON CONDITIONS FOR MATRICES T SUCH THAT T-I AND I-T^(-1) ARE INVERSE H-MATRICES*

Gormantara, Jeriko
Garminia, Hanni
Amir, Amir Kamal
Ramdan, Evan

BAREKENG: Jurnal Ilmu Matematika dan Terapan
Vol 19 No 4 (2025): BAREKENG: Journal of Mathematics and Its Application

In this article, we study an analogue of a classical result for M-matrices: if T - I is an invertible M-matrix, then both T and I - T-1 are also invertible M-matrices. We extend this implication to a broader class—inverse H-matrices. The expressions T - I and I - T-1 commonly arise in the analysis of matrix stability, convergence of iterative methods, and spectral transformations, making their structural properties important for numerical analysis. We demonstrate that this implication does not generally hold for inverse H-matrices. However, we derive some conditions under which it remains valid. Specifically, we prove that under certain conditions, if T - I is an inverse H-matrix, then T and I - T-1 are also inverse H-matrices. Additionally, we investigate the result in the context of group inverses, showing that it does not hold for group inverse M-matrices and H-matrices.