Given a set of full-rank matrices , this brief paper proposes a method based on linear feasibility tests to determine whether a convex combination , with in the unit simplex , may result in a rank-deficient matrix. The method is based on a sequence of linear programs with increasingly tightened constraints, and is guaranteed to reach an outcome after a finite number of iterations. Given a tolerance arbitrarily chosen by the user, the method will either (i) certify that such that is rank-deficient or (ii) yield , such that , which certifies that the smallest singular value of is less than . This method bridges a gap in the literature, as no other numerically verifiable test for generic , , has been proposed to reach the conclusion (ii). Three numerical examples are provided to showcase the advantages of the proposed method with respect to other tests reported in previous papers. The code employed in this work is available at .

中文翻译：

---

一种基于线性可行性检验的矩阵凸组合满秩表征方法


给定一组满秩矩阵，这篇简短的论文提出了一种基于线性可行性测试的方法，以确定单位单纯形的凸组合是否可能产生秩亏矩阵。该方法基于一系列具有日益严格的约束的线性程序，并保证在有限次数的迭代后达到结果。给定用户任意选择的容差，该方法将 (i) 证明 是秩不足的，或者 (ii) 产量 ，使得 ，证明 的最小奇异值小于 。该方法弥补了文献中的空白，因为没有提出其他可对通用 , , 进行数值验证的测试来得出结论 (ii)。提供了三个数值示例来展示所提出的方法相对于先前论文中报告的其他测试的优势。这项工作中使用的代码可在 处获得。