Stone猜想的部分进展：带正行列式的完全半单调矩阵的$P_0$成员性
Partial Progress on Stone's Conjecture: $P_0$-Membership of Fully Semimonotone Matrices with Positive Determinant
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中文总结 AI 辅助
本文证明任意阶带正行列式的完全半单调矩阵$E_0^f$均为$P_0$矩阵，同时构造反例说明该正行列式假设无法由$Q_0$成员性推出，Stone猜想仍未解决。
中文摘要 AI 辅助
Stone（斯坦福大学运筹学系博士论文，1981）证明，$U \bigcap Q_0$中的每个矩阵都是$P_0$矩阵，并猜想更大类的完全半单调$Q_0$矩阵$E_0^f \bigcap Q_0$也满足同一结论。Murthy和Parthasarathy[《SIAM矩阵分析与应用杂志》16卷（1995），1268-1286]验证了阶数不超过$4 \times 4$的矩阵的该猜想，在附加假设下验证了$5 \times 5$和$6 \times 6$矩阵的该猜想，还验证了任意阶数的几个特殊子类的该猜想，但该猜想整体仍未解决。本文证明，对任意阶数$n$的矩阵，每个带正行列式的$E_0^f$矩阵都是$P_0$矩阵；我们的证明通过对$n$进行归纳，利用主枢轴变换下的主子式代数分析完成。我们进一步展示了一个属于$E_0^f$且$\boldsymbol{\text{det } A > 0}$但不属于$Q_0$的矩阵$A$，表明我们定理中使用的$\text{det } A > 0$这一假设本身无法从$Q_0$成员性推导得出，因此仅靠该假设无法证明Stone猜想。Stone猜想本身仍未解决。
英文摘要
Stone (Ph.D.\ thesis, Department of Operations Research, Stanford University, 1981) proved that every matrix in $U \cap Q_0$ is a $P_0$-matrix and conjectured that the same conclusion holds for the larger class $E_0^f \cap Q_0$ of fully semimonotone $Q_0$-matrices. Murthy and Parthasarathy [SIAM J.\ Matrix Anal.\ Appl.\ 16 (1995), 1268--1286] verified the conjecture for matrices of order up to $4 \times 4$, for $5 \times 5$ and $6 \times 6$ matrices under additional hypotheses, and for several special subclasses of arbitrary order, but the conjecture remains open in general. In this paper we prove that every $E_0^f$-matrix with positive determinant is a $P_0$-matrix, for matrices of arbitrary order $n$; our proof proceeds by induction on $n$, via an algebraic analysis of principal minors under principal pivotal transforms. We further exhibit a matrix $A \in E_0^f$ with $\det A > 0$ that fails to belong to $Q_0$, showing that the hypothesis $\det A > 0$ used in our theorem cannot, by itself, be deduced from membership in $Q_0$, and hence does not on its own yield a proof of Stone's conjecture. Stone's conjecture itself remains open.
