Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of $\mathbf{M}$- and $\mathbf{H}$-Matrices and Their Inverses

Abstract

The quantity $q(A\circ A^{-1})$, the minimum modulus of the eigenvalues of $A\circ A^{-1}$, arises naturally in connection with positive diagonal symmetrizability. For an invertible $\mathbf{M}$-matrix $A$ of order $n$, the classical bounds $\frac{2}{n}\leq q(A\circ A^{-1})\leq 1$ are known. We discuss the sharpness of the lower bound $\frac{2}{n}$ and investigate the converse of a related result involving the Jacobi iteration matrix. In particular, we show that $\rho(J_{A_k})\to 1$ does not, in general, imply $q(A_k\circ A_k^{-1})\to \frac{2}{n}$, and identify a class for which this implication holds. We then turn to invertible $\mathbf{H}$-matrices, a broader class that contains invertible $\mathbf{M}$-matrices. We show that $A\circ A^{-1}$ is an invertible $\mathbf{H}$-matrix whenever $A$ is an invertible $\mathbf{H}$-matrix. In contrast to the $\mathbf{M}$-matrix setting, $q(A\circ A^{-1})$ can be arbitrarily close to zero. However, replacing $A^{-1}$ by the inverse of the comparison matrix restores the classical lower bound: we prove that $q(A\circ\mathcal{M}(A)^{-1})\geq \frac{2}{n}$ and obtain further bounds involving the Jacobi iteration matrix of $\mathcal{M}(A)$. Finally, for positive diagonally symmetrizable invertible $\mathbf{H}$-matrices, we establish the upper bound $q(A\circ A^{-1})\leq1$ and, in the irreducible case, characterize when equality occurs.