Gram-Schmidt Orthonormalization

Orthonormalize a set of vectors

qₖ = (vₖ − Σⱼ ⟨vₖ, qⱼ⟩ qⱼ) / ‖vₖ − Σⱼ ⟨vₖ, qⱼ⟩ qⱼ‖

Orthonormal basis

q₁ = (0.7071, 0.7071, 0)

q₂ = (0.4082, -0.4082, 0.8165)

q₃ = (-0.5774, 0.5774, 0.5774)

Independent vectors

3 of 3

Orthogonality residual

≈ 0

Verified orthonormal

Gram-Schmidt process

v₁ = (1, 1, 0)
Length ‖u₁‖ = 1.4142
q₁ = (0.7071, 0.7071, 0)
v₂ = (1, 0, 1)
− 0.7071·q₁
Length ‖u₂‖ = 1.2247
q₂ = (0.4082, -0.4082, 0.8165)
v₃ = (0, 1, 1)
− 0.7071·q₁ − 0.4082·q₂
Length ‖u₃‖ = 1.1547
q₃ = (-0.5774, 0.5774, 0.5774)