发表机构
Indian Institute of Technology Gandhinagar; University of Regina; Indian Institute of Technology Patna（印度理工学院甘地纳加尔分校; 里贾纳大学; 印度理工学院巴特那分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究M-矩阵与H-矩阵及其逆的Hadamard积的最小特征值模的界，探讨经典下界的尖锐性，证明H-矩阵情形下下界可任意小，但用比较矩阵逆可恢复下界，并给出上界及等号条件。
AI 中文摘要
量$q(A\circ A^{-1})$，即$A\circ A^{-1}$的特征值的最小模，自然地出现在正对角对称化问题中。对于$n$阶可逆$\mathbf{M}$-矩阵$A$，已知经典界$\frac{2}{n}\leq q(A\circ A^{-1})\leq 1$。我们讨论下界$\frac{2}{n}$的尖锐性，并研究涉及Jacobi迭代矩阵的相关结果的逆命题。特别地，我们证明$\rho(J_{A_k})\to 1$一般并不蕴含$q(A_k\circ A_k^{-1})\to \frac{2}{n}$，并确定了一类使该蕴含成立的矩阵类。然后我们转向可逆$\mathbf{H}$-矩阵，这是包含可逆$\mathbf{M}$-矩阵的更广的矩阵类。我们证明当$A$是可逆$\mathbf{H}$-矩阵时，$A\circ A^{-1}$也是可逆$\mathbf{H}$-矩阵。与$\mathbf{M}$-矩阵情形相反，$q(A\circ A^{-1})$可以任意接近零。然而，将$A^{-1}$替换为比较矩阵的逆可恢复经典下界：我们证明$q(A\circ\mathcal{M}(A)^{-1})\geq \frac{2}{n}$，并得到涉及$\mathcal{M}(A)$的Jacobi迭代矩阵的进一步界。最后，对于正对角对称化的可逆$\mathbf{H}$-矩阵，我们建立上界$q(A\circ A^{-1})\leq 1$，并在不可约情形下刻画等号成立的条件。
英文摘要
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 $ρ(J_{A_k})\to 1$ 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.
Comments19 pages
