QR Factorization Through Direct Plane Rotations

This preprint describes a QR factorization in which each active column is transformed into its coordinate axis by a single direct plane rotation. The rotation acts in the plane spanned by the active column and the corresponding coordinate vector while leaving the orthogonal complement unchanged. Recursive deflation produces a QR factorization, and a block formulation is derived that allows the transformation to be applied without explicitly constructing the full rotation matrix. The underlying arbitrary-dimensional plane rotation is known. This note focuses on its recursive use for QR factorization and on the resulting computational formulas.