Linear least squares solutions by householder transformations

TLDR: In this paper, the euclidean norm is unitarily invariant and a vector x is determined such that x is parallel b-Ax parallel = \parallel c - QAx parallel where c denotes the first n components of c.

Abstract: Let A be a given m×n real matrix with m≧n and of rank n and b a given vector. We wish to determine a vector x such that $$\parallel b - A\hat x\parallel = \min .$$ where ∥ … ∥ indicates the euclidean norm. Since the euclidean norm is unitarily invariant $$\parallel b - Ax\parallel = \parallel c - QAx\parallel $$ where c=Q b and Q T Q = I. We choose Q so that $$QA = R = {\left( {_{\dddot 0}^{\tilde R}} \right)_{\} (m - n) \times n}}$$ (1) and R is an upper triangular matrix. Clearly, $$\hat x = {\tilde R^{ - 1}}\tilde c$$ where c denotes the first n components of c.