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.

本研究解决了在高斯噪声影响下，正交矩阵条件数变化的问题，特别针对在QR分解算法中正交性不完美的情况。通过推导在无完美正交假设下矩阵条件数增加的界限，我们得出了一系列关于最小二乘法和正交投影性能的新结果，推动了理论的发展。

高斯噪声下最小二乘法、正交投影和QR分解算法的概率分析