QRF, Gram-Schmidt, part 3

Gram-Schmidt can be understood as a sequence of localized geometric updates that change one column at a time.

• Each subsequent matrix is obtained by right-multiplication Aₖ₊₁ = Aₖ Sₖ Dₖ, so one active column is changed at each step
• The shear matrices remove overlap with earlier orthogonal directions, while the diagonal matrices normalize the remaining column
• Q is produced by gradually removing scale and shear from A
• R accumulates the removed scale and shear through the inverse right-side updates
• The upper-triangular shape of R follows from the upper-triangular shape of every Sₖ, Dₖ and their inverses

This stepwise viewpoint makes QR factorization look like a sequence of transformations rather than a single final identity.

Because the usual symbolic formulas hide the fact that Gram-Schmidt changes one column at a time through specific geometric operations. The stepwise view shows exactly what is being removed, what is being normalized and how those updates accumulate into Q and R.

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Because Gram-Schmidt changes columns, and right-multiplication is the natural way to update columns one at a time. Each shear or scaling matrix differs from the identity in one active column, so the sequence Aₖ₊₁ = Aₖ Sₖ Dₖ tracks the orthogonalization column by column.

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