For an $$m \times n$$ matrix A, the mathematical property that the rank of A is equal to r for $$0< r < \min (m,n)$$ is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer k and a real or complex matrix A, square or rectangular, we first present a verification algorithm to compute a narrow interval matrix $$\varDelta $$ with the property that there exists a matrix within $$A-\varDelta $$ with at least rank deficiency k. Subsequently, we extend this algorithm for computing an inclusion for a specific perturbation E with that property but also a minimal distance with respect to any unitarily invariant norm. For this purpose, we generalize Wedin’s $$\sin (\theta )$$ theorem by removing its orthogonality assumption. The corresponding result is the singular vector space counterpart to Davis and Kahan’s generalized $$\sin (\theta )$$ theorem for eigenspaces. The presented verification methods use only standard floating-point operations and are completely rigorous including all possible rounding errors and/or data dependencies.

中文翻译：

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通过 Wedin 的 $$\sin (\theta )$$ 定理的推广，验证了指定秩不足的最近矩阵的包含

对于 $$m \times n$$ 矩阵 A，对于 $$0< r < \min (m,n)$$ A 的秩等于 r 的数学性质是不适定问题。在本笔记中，我们表明，无论这种情况如何，都可以以数学上严格的方式并仅使用浮点运算来解决计算具有至少秩不足 k 的邻近矩阵的强相关问题。给定一个整数 k 和一个实数或复数矩阵 A，正方形或矩形，我们首先提出一个验证算法来计算一个窄区间矩阵 $$\varDelta $$，它的性质是 $$A-\varDelta $ 中存在一个矩阵$ 至少有秩不足 k。随后，我们扩展了该算法，用于计算具有该属性的特定扰动 E 的包含，但也是相对于任何单一不变范数的最小距离。为此，我们通过去除其正交性假设来概括 Wedin 的 $$\sin (\theta )$$ 定理。相应的结果是对应于 Davis 和 Kahan 的广义 $$\sin (\theta )$$ 特征空间定理的奇异向量空间。所提出的验证方法仅使用标准浮点运算，并且是完全严格的，包括所有可能的舍入误差和/或数据相关性。