About Gram Schmidt Orthonormalization Calculator (Formula)
The Gram-Schmidt Orthonormalization Calculator is a mathematical tool used in linear algebra to transform a set of linearly independent vectors into an orthonormal basis. This process involves creating a new set of vectors that are orthogonal (perpendicular) to each other and have a magnitude of 1 (unit vectors). The Gram-Schmidt orthonormalization process is typically applied to a set of vectors {v₁, v₂, v₃, …} to obtain a new set of orthonormal vectors {u₁, u₂, u₃, …}.
The formula for the Gram-Schmidt process is a recursive process that can be summarized as follows:
- Start with the first vector u₁, which is simply the normalized version (unit vector) of the first vector v₁: u₁ = v₁ / ||v₁||, where ||v₁|| represents the magnitude (length) of v₁.
- For each subsequent vector vᵢ, where i > 1, calculate the new orthonormal vector uᵢ by subtracting the projections of vᵢ onto the previously determined orthonormal vectors u₁, u₂, …, uᵢ₋₁ from vᵢ, and then normalize the result to obtain uᵢ.
The recursive formula for uᵢ can be expressed as: uᵢ = (vᵢ – (vᵢ ⋅ u₁)u₁ – (vᵢ ⋅ u₂)u₂ – … – (vᵢ ⋅ uᵢ₋₁)uᵢ₋₁) / ||(vᵢ – (vᵢ ⋅ u₁)u₁ – (vᵢ ⋅ u₂)u₂ – … – (vᵢ ⋅ uᵢ₋₁)uᵢ₋₁)||
Where:
- uᵢ represents the orthonormal vector being calculated.
- vᵢ is the current vector being orthonormalized.
- u₁, u₂, …, uᵢ₋₁ are the previously calculated orthonormal vectors.
- “⋅” represents the dot product of vectors.
- ||…|| denotes the magnitude or length of a vector.
The Gram-Schmidt process transforms the set of vectors into an orthonormal basis, which is valuable in various mathematical and engineering applications, including solving linear equations, computing projections, and performing eigenvalue decompositions.