Perfectly Normal

LEARNING GOALS

Choose the next Gram–Schmidt move.

A = [a₁ ⋯ aₙ], independent columns; for j = 1,…,n:

1 Choose column Add the current original column aⱼ to the inputs.

2 Project For i < j: compute rᵢⱼ = ⟨qᵢ, aⱼ⟩, then build pⱼ = Σᵢ<ⱼrᵢⱼqᵢ.

3 Residual vⱼ = aⱼ − pⱼ; then vⱼ ⟂ span{q₁,…,qⱼ₋₁}.

4 Measure rⱼⱼ = ‖vⱼ‖ > 0.

5 Normalize qⱼ = vⱼ/rⱼⱼ.

QR Collect Q = [q₁ ⋯ qₙ], R = (rᵢⱼ) upper triangular; QᵀQ = I, A = QR.