最小与最大矩阵锥上的按元素Loewner保持映射
Entrywise Loewner Preservers on Min and Max Matrix Cones
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中文总结 AI 辅助
本文刻画了Min与Max矩阵锥上按元素Loewner保持映射的性质，明确了函数保持半正定性、全非负性等的条件，还建立了严格锥与相关矩阵类的关联。
中文摘要 AI 辅助
设 \\(x=(x_1,\ldots,x_n)\\) 为实序列，由其生成的最小矩阵 \\(A_{\min}(x)=(x_{\min(i,j)})_{i,j=1}^n\\) 和最大矩阵 \\(A_{\max}(x)=(x_{\max(i,j)})_{i,j=1}^n\\) 分别对应Min矩阵与Max矩阵。利用这两类矩阵锥的经典参数化方法，本文给出了按元素映射保持半正定性、全非负性、Loewner序及Loewner凸性的精确刻画。核心结果围绕Loewner序展开：假设连续性或可微性，按元素Loewner序保持等价于函数 \\(f\\) 非减且凸；在Min与Max锥上，要求对任意元素对满足Loewner凸性不等式的条件过强，会迫使 \\(f\\) 为仿射函数；若按标准约定将不等式限制在Loewner可比的元素对上，该条件会自动要求 \\(f\in C^1([0,\infty))\\)，且等价于 \\(f\\) 与 \\(f'\\) 均为凸函数，Loewner凹性也有类似结论，每种情况的条件均可在二维维度中检测。此外，本文证明函数 \\(f:[0,\infty)\to\mathbb R\\) 按元素保持所有正半定Min或Max矩阵的半正定性当且仅当 \\(f\\) 非负且非减，该条件同时刻画了全非负性的保持性；最后，本文确定了幂函数的作用范围与Loewner序自同构，刻画了严格Min与严格Max类的按元素保持映射，并将这些严格锥与逆M-矩阵、振荡矩阵建立关联。
英文摘要
Let \[ A_{\min}(x)=\bigl(x_{\min(i,j)}\bigr)_{i,j=1}^n, \qquad A_{\max}(x)=\bigl(x_{\max(i,j)}\bigr)_{i,j=1}^n \] be the Min and Max matrices generated by a real sequence \(x=(x_1,\ldots,x_n)\). Using their classical cone parametrizations, we give exact characterizations of entrywise maps preserving positive semidefiniteness, total nonnegativity, Loewner order, and Loewner convexity. Our main results concern the Loewner structure.Without assuming continuity or differentiability, entrywise Loewner-order preservation is equivalent to \(f\) being nondecreasing and convex. On the Min and Max cones, requiring the Loewner-convexity inequality on arbitrary pairs is rigid and forces \(f\) to be affine. Under the standard convention of restricting the inequality to Loewner-comparable pairs, the condition automatically forces \(f\in C^1([0,\infty))\) and is equivalent to convexity of both \(f\) and \(f'\). The analogous statement holds for Loewner concavity. In each case, the condition is already detected in dimension two. We also show that \(f:[0,\infty)\to\mathbb R\) preserves positive semidefiniteness entrywise on all positive semidefinite Min or Max matrices if and only if \(f\) is nonnegative and nondecreasing; the same condition characterizes total-nonnegativity preservation. Finally, we determine the power-function ranges and Loewner-order automorphisms, characterize the entrywise preservers of the strict Min and Max classes, and relate these strict cones to inverse \(M\)-matrices and oscillatory matrices.
