A Gram-Schmidt Calculator is a mathematical tool used to perform the Gram-Schmidt orthogonalization process.
This process transforms a set of linearly independent vectors into an orthogonal or orthonormal set of vectors.
Gram-Schmidt Example Chart
| Step | Vector | Calculation | Result |
|---|---|---|---|
| 1 | u₁ | v₁ | (1, 1, 0) |
| 2 | e₁ | u₁ / ‖u₁‖ | (1/√2, 1/√2, 0) |
| 3 | u₂ | v₂ – proj_u₁(v₂) | (-1, 1, 0) |
| 4 | e₂ | u₂ / ‖u₂‖ | (-1/√2, 1/√2, 0) |
| 5 | u₃ | v₃ – proj_u₁(v₃) – proj_u₂(v₃) | (0, 0, 2) |
| 6 | e₃ | u₃ / ‖u₃‖ | (0, 0, 1) |
Gram-Schmidt Formula
The Gram-Schmidt Formula for orthogonalization is:
- u₁ = v₁
- u₂ = v₂ – proj_u₁(v₂)
- u₃ = v₃ – proj_u₁(v₃) – proj_u₂(v₃) …
Where proj_u(v) is the projection of v onto u, given by:
proj_u(v) = (v · u / u · u) * uLet’s orthogonalize the vectors v₁ = (1, 1, 0), v₂ = (0, 1, 0), and v₃ = (0, 1, 1).
- u₁ = v₁ = (1, 1, 0)
- u₂ = v₂ – proj_u₁(v₂) = (0, 1, 0) – ((0, 1, 0) · (1, 1, 0) / (1, 1, 0) · (1, 1, 0)) (1, 1, 0) = (0, 1, 0) – (1/2) (1, 1, 0) = (-1/2, 1/2, 0)
- u₃ = v₃ – proj_u₁(v₃) – proj_u₂(v₃) = (0, 1, 1) – (1/2) (1, 1, 0) – (1/3) (-1/2, 1/2, 0) = (-1/2, 1/6, 1)
The resulting orthogonal set is {(1, 1, 0), (-1/2, 1/2, 0), (-1/2, 1/6, 1)}.
Gram-Schmidt Process
The Gram-Schmidt process takes a finite, linearly independent set of vectors S = {v₁, …, vₖ} and generates an orthogonal set S’ = {u₁, …, uₖ} that spans the same k-dimensional subspace of Rⁿ as S.
The process works as follows:
- Start with the first vector: u₁ = v₁
- For each subsequent vector vᵢ (i = 2, …, k): a. Subtract from vᵢ its projection onto each of the previous orthogonalized vectors uj (j = 1, …, i-1) b. The result is the next orthogonal vector uᵢ
Mathematically, this is expressed as:
uᵢ = vᵢ - Σ(j=1 to i-1) proj_uj(vᵢ)Where proj_u(v) = (v · u / u · u) * u
Let’s apply the Gram-Schmidt process to the vectors v₁ = (1, 0, 1), v₂ = (1, 1, 0), and v₃ = (1, 1, 1) in R³.
- u₁ = v₁ = (1, 0, 1)
- u₂ = v₂ – proj_u₁(v₂) = (1, 1, 0) – ((1, 1, 0) · (1, 0, 1) / (1, 0, 1) · (1, 0, 1)) (1, 0, 1) = (1, 1, 0) – (1/2) (1, 0, 1) = (1/2, 1, -1/2)
- u₃ = v₃ – proj_u₁(v₃) – proj_u₂(v₃) = (1, 1, 1) – (2/2) (1, 0, 1) – (1/3) (1/2, 1, -1/2) = (0, 1, 0) – (1/6, 1/3, -1/6) = (-1/6, 2/3, 1/6)
The resulting orthogonal set is {(1, 0, 1), (1/2, 1, -1/2), (-1/6, 2/3, 1/6)}.