Upper bound for real invertible H-matrices

Determine whether every real invertible H-matrix A satisfies the upper bound q(A∘A^{-1})≤1, and, if not, characterize the subclasses of real invertible H-matrices for which this inequality holds.

Background

The paper establishes q(A∘A{-1})≤1 for 2×2 invertible H-matrices and for positive diagonally symmetrizable invertible H-matrices of arbitrary order. In the irreducible positive diagonally symmetrizable case, it also characterizes equality: q(A∘A{-1})=1 exactly when all diagonal entries of A have the same sign.

The authors do not resolve whether the same upper bound holds for every real invertible H-matrix. They note that the question is genuinely field-dependent: a complex invertible H-matrix counterexample is given with q(A∘A{-1}) approximately 1.09552>1. Thus, the unresolved issue is specific to the real setting and asks either for a proof of the universal bound or for a characterization of the real cases in which it remains valid.

References

While the inequality $q(A\circ A{-1})\leq1$ remains an open question for general invertible $\mathbf{H}$-matrices with real entries, it does not hold in general over the complex field.

— Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of $\mathbf{M}$- and $\mathbf{H}$-Matrices and Their Inverses  (2609.24019 - Chauhan et al., 21 Sep 2026) in Section Conclusion; reiterated in Section Open Problems and Future Directions