The Equality Cases for the Inequalities of Fischer, Oppenheim, and Ando for General M-Matrices

Abstract

In this paper, we study the zero pattern structure of two general M-matrices whose Fan product is singular. Then, for general M-matrices, necessary and sufficient conditions are obtained for equality in the inequalities of Fischer, Oppenheim, and Ando.

An Oppenheim type inequality for positive definite block matrices

“…Over the years, various generalizations and extensions of (4) and ( 5) have been obtained in the literature. For instance, see [23,24] for the equality cases; see [2,15,22,4,7] for the extensions of M -matrices. It is worth noting that Lin [16] recently gave a remarkable extension (Theorem 1.1) of Chen's result (5) for positive definite block matrices.…”

An Oppenheim type determinantal inequality for the Khatri-Rao product

“…Over the past years, various generalizations and extensions of (4) and ( 5) have been obtained in the literature. For instance, see [18,19] for the equality cases; see [1,10,17,3] for the extensions of M -matrices; see [6,14,4] for the extensions of block Hadamard product.…”

The roots and links in a class of $M$-matrices

“…On the inequality of M− matrices and inverse M− matrices, Ando in [1] presents many nice and excellent inequalities which play an key role in the nonnegative matrix theory. Zhang [15] characterized equality cases in Fisher, Oppenheim and Ando inequalities. For more detail information on inverse M− and Z− matrices, the reader is referred to [6], [8], [9], [10], [16] and the references in there.…”