When modified Gram-Schmidt generates a well-conditioned set of vectors

In this paper, we show why the modified Gram–Schmidt algorithmgenerates a well-conditioned set of vectors. This result holdsunder the assumption that the initial matrix is not ‘tooill-conditioned’ in a way that is quantified. As a consequencewe show that if two iterations of the algorithm are performed,the resulting algorithm produces a matrix whose columns areorthogonal up to machine precision. Finally, we illustrate througha numerical experiment the sharpness of our result.