In this paper, we extend the work of Liesen et al. (2002), which analyzes how the condition number of an orthonormal matrix Q changes when a column is added ([Q, c]), particularly focusing on the perpendicularity of c to the span of Q. Their result, presented in Theorem 2.3 of Liesen et al. (2002), assumes exact arithmetic and orthonormality of Q, which is a strong assumption when applying these results to numerical methods such as QR factorization algorithms. In our work, we address this gap by deriving bounds on the condition number increase for a matrix B without assuming perfect orthonormality, even when a column is not perfectly orthogonal to the span of B. This framework allows us to analyze QR factorization methods where orthogonalization is imperfect and subject to Gaussian noise. We also provide results on the performance of orthogonal projection and least squares under Gaussian noise, further supporting the development of this theory.

翻译：本文拓展了Liesen等人（2002）的研究工作，该研究分析了当向标准正交矩阵Q添加列向量c形成增广矩阵[Q, c]时，其条件数如何变化，尤其关注c与Q张成空间的正交性。Liesen等人（2002）定理2.3中呈现的结果基于精确算术与Q的标准正交性假设，这一强假设在将结论应用于QR分解等数值方法时具有局限性。本研究通过建立矩阵B条件数增长的上界来弥补这一不足，且不要求B具有完全的标准正交性，即使新增列向量与B的张成空间不完全正交。该理论框架使我们能够分析正交化过程不完美且受高斯噪声影响的QR分解方法。我们同时给出了高斯噪声下正交投影与最小二乘法性能的相关结果，进一步支撑了该理论体系的发展。