发表机构
Hefei University of Technology（合肥工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限仿射群秩一轨道矩阵恢复，通过平方和恒等式与块分解，确定显式实窗口的最小奇异值下界，构造唯一最优三级迹窗口，并证明其条件数下界大于1/√2。
AI 中文摘要
设 \\(q=p^h\\) 为奇素数幂，仿射群 \\(\\(\mathbb F_q\rtimes\mathbb F_q^\times\\)\\) 通过其典型的 \\((q-1)\\) 维不可约表示作用。这些秩一轨道的定性矩阵恢复是已知的。我们为显式实生成窗口确定了锐利的下奇异值界。首先，我们计算了 \\(h\geq2\\) 时 \\(\mathbb F_{p^h}\\) 上每个非负两级窗口的精确最小奇异值，并在 \\(q-1\geq10\\) 时确定了唯一的最优窗口。所得条件数在每个固定的奇特征塔上保持远离零，并且与所有实窗口上的最佳可能值相差一个显式的依赖于特征的因子。对于每个 \\(q=3^h\\)，\\(h\geq2\\)，我们构造了一个实三级绝对迹窗口（在 \\(\operatorname{Tr}_{\mathbb F_q/\mathbb F_3}\\) 的纤维上为常数），其最小奇异值为 \\[ \frac{q(\sqrt2-1)}{q(2-\sqrt2)-1}>\frac1{\sqrt2}. \\] 对于在三个迹类上为常数的完整实窗口类，我们将最小奇异值化简为四个标量表达式和一个对称的 \\(2\times2\\) 矩阵。这给出了精确的全局最优值和所有等式情形：所展示的迹窗口在全局符号和两个非零迹类的互换意义下是唯一最优的。因此，零迹均值由最优性得出。证明结合了显式的平方和恒等式、一致的二次型证书以及轨道测量算子的有限几何块分解。
英文摘要
Let \(q=p^h\) be an odd prime power, and let the affine group \(\mathbb F_q\rtimes\mathbb F_q^\times\) act through its canonical \((q-1)\)-dimensional irreducible representation. Qualitative matrix recovery for these rank-one orbits is known. We determine sharp lower singular-value bounds for explicit real generating windows. First, we compute the exact least singular value for every nonnegative two-level window over \(\mathbb F_{p^h}\) with \(h\geq2\), and identify the unique optimizer when \(q-1\geq10\). The resulting conditioning stays bounded away from zero on every fixed odd-characteristic tower and is within an explicit characteristic-dependent factor of the best possible value over all real windows. For every \(q=3^h\), \(h\geq2\), we construct a real three-level absolute-trace window (constant on the fibers of \(\operatorname{Tr}_{\mathbb F_q/\mathbb F_3}\)) whose least singular value is \[ \frac{q(\sqrt2-1)}{q(2-\sqrt2)-1}>\frac1{\sqrt2}. \] For the full class of real windows constant on the three trace classes, we reduce the least singular value to four scalar expressions and a symmetric \(2\times2\) matrix. This yields the exact global optimum and all equality cases: the displayed trace window is uniquely optimal up to global sign and interchange of the two nonzero trace classes. Thus zero trace mean follows from optimality. The proof combines an explicit sum-of-squares identity, uniform quadratic-form certificates, and a finite-geometric block decomposition of the orbit measurement operator.
