This chapter develops the structure of matrix R, shows why upper-triangular form appears in QR factorization and explains how QR separates deformation from rotation or reflection. It then presents geometric views of the factorization in square and tall cases, interprets the hierarchy encoded by R and ends with the uniqueness of thin QR factorization.
After Gram-Schmidt produces Q, the matrix R records how the original columns are assembled from the orthonormal directions. Its upper-triangular form reveals both algebraic dependence and geometric order.
Together with part 1, this chapter completes the structural and geometric picture of QR factorization.
Because each orthogonal direction is built from the current column after removing components along earlier directions. That makes later basis vectors orthogonal to earlier columns, which creates zeros below the diagonal in QᵀA = R.
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Each orthogonal direction could be flipped in sign, but requiring nonnegative diagonal entries in R removes that ambiguity. Under that convention, thin QR gives a unique orthonormal basis for the column space together with a unique upper-triangular factor.