Gram-Schmidt Process

Gram-Schmidt Process

Wanho Choi (wanochoi.com)

Gram-Schmidt Process

• Input:

Non-orthogonal set of independent vectors

• _Output:

Orthogonal set of vectors

It takes a linearly independent vector set for and generates an orthogonal set

that spans the same -dimensional subspace of as .

k S = {v₁, ⋯, v_k} k ≤ n

S′ = {u₁, ⋯, u_k}

k IRn S

{v_i}

Gram-Schmidt Process

v v₂

Gram-Schmidt Process

v = u

Gram-Schmidt Process

v = u

Gram-Schmidt Process

v₁ = u₁ v₂

e₁ = u1 u

Gram-Schmidt Process v₁ = u₁ v₂ e₁ = u1 u e₁ ⋅v₂

Gram-Schmidt Process v₁ = u₁ v₂ e₁ = u1 u e₁ ⋅v₂ = proj_u1

( )

Gram-Schmidt Process v₁ = u₁ v₂ e₁ = u1 u e₁ ⋅v₂ = proj_u1

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( )

Gram-Schmidt Process v₁ = u₁ v₂ e₁ = u1 u e₁ ⋅v₂ = proj_u1

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Gram-Schmidt Process u₂ = v₂ − proj_u 1

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∑

Gram-Schmidt Process u₂ = v₂ − proj_u 1

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∑

Gram-Schmidt Process u₂ = v₂ − proj_u 1

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∑