Gram Schmidt Orthonormalization Process With an Example

In linear algebra and numerical analysis, Gram–Schmidt orthonormalization process is a process of transforming two or more vectors that are perpendicular to each other.

• It is a case of inner product space, which is characterized by a set of linearly independent vectors.
• The Gram-Schmidt orthonormalization process is also known as the orthogonalization or the Gram-Schmidt algorithm.
• It is a set of linearly independent vectors S = {v₁, ..., v_k} for k ≤ n and generates an orthogonal set S′ = {u₁, ..., u_k}.
• The process is useful for computing an orthonormal basis in an inner product space.
• The Gram-Schmidt orthonormalization process, a fundamental tool in mathematical analysis, finds its applications in various real-world scenarios. 
• It is used in motion planning, sensor calibration, and robot localization tasks, demonstrating its practical significance beyond theoretical understanding.

Key Terms: Gram-Schmidt Orthonormalization Process, Vectors, Linear Algebra, Euclidean Space, Variables

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What is Gram-Schmidt Orthonormalization Process?

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The Gram-Schmidt orthonormalization process comprises a series of steps that transform a set of linearly independent vectors into a set of orthonormal vectors that cover the same space as the original set.

• It is most commonly used in the Euclidean space Rn, which is equipped with the standard inner product.
• The Gram-Schmidt orthonormalization process was discovered by Jørgen Pedersen Gram and Erhard Schmidt.
• The process takes into account those vectors that generate the same number of orthonormal vectors.
• The method was generalized by the Iwasawa decomposition using the theory of lie group decompositions.
• It is used in matrices where column vectors of a rank matrix yield an orthogonal and a triangular matrix.

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State and Prove Gram-Schmidt Orthonormalization Process

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Statement: Let (v₁,...,v_m) be a set of linearly independent vectors in V lying in the k-dimensional subspace of R_n then (e₁,...,e_m) be an orthonormal list.

Proof: We will begin with any basis for vector V to determine the orthonormal basis for vector V.

= Consider {v₁,…,v_k} to be a non-orthonormal matrix for V. 

= We will form a set of {u₁,…,u_k} repeatedly until {u₁,…,u_p} will form an orthonormal basis for the vector {v₁,…,v_p}.

⇒ For p = 1 Let u₁=1/ ∥v∥ 

= Suppose {u₁,…,u_p-1} is an orthonormal basis for the span of a vector set {v₁,…,v_p}.

= Now determine the vector d such that ‖d‖=1,ui⋅d=0 for i=1,…,p−1.

= vp can be represented as a linear combination of u₁,…,u_p−1,d.

⇒ For time being eliminate the condition ‖d‖=1 and determine the value for d in the following system:

u₁. d = 0

u₂.d = 0

…

…

…

u_p-1.d = 0

= λ₁u₁⋯⋅⋯λ_p−1u_p−1+αd = v_p

= λ₁,…,λ_p−1 and d₁,…,d_n represent the set of variables here. However, the final equation used in the system is not linear. 

Since vp does not lie inside the range of u₁,…,u_p−1, we cannot have any solution for value α=0. 

= Now consider that α≠0 and solve the system of equations by substituting the value of u=αd.

(1/α)u₁⋅u=0

(1/α)u₂⋅u=0

…

…

…

(1/α)u_p-1⋅u=0

λ₁u₁⋯⋅⋯λp₋₁u_p−1+u = v_p

= In other words it can be represented as:

u₁⋅u = 0

u₂⋅u = 0

u_p-1⋅u = 0

= λ₁u₁⋯⋅⋯λp₋₁u_p−1+u = v_p

= Now determine u = vp−(λ₁u₁+⋯+λ_p−1u_p−1) using the last equation of the system. 

= Substituting the ith value in the equation i∈{1,…,p−1}, we will get

ui ⋅ (vp−(λ₁u₁+⋯+λ_p−1u_p−1))=0

= This is equivalent to ui⋅v_p=λ_i. 

= The required value of u is represented as: u = vp−((u₁⋅v_p)u₁ + (u₂⋅vp)u₂ + ⋯ +(u_p−1⋅vp)u_p−1) where u≠0. 

= Now consider up=(1 / ∥u∥)u.

⇒ The summary of the procedure is as follows:

u₁=(1/||v₁||). v₁

u₂ = (1/||e₂||). e₂ where e₂=v₂−(u₁⋅v₂)u₁

u₃ = (1/||e₃||). e₃ where e₃=v₃−((u₁⋅v₃)u₁+(u₂⋅v₃)u₂)

…

…

u_i = (1/||e_i||). e_i, where e_i=v_i−((u₁⋅v_i)u₁+(u₂⋅v_i)u₂+⋯+(u_i−1⋅v_i)u_i−1))

…

…

…

u_k = (1/||e_k||). e_k where e_k=v_k−((u₁⋅vk)u₁+(u₂⋅vk)u₂+⋯+(u_k−1⋅vk)u_k−1))

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Gram-Schmidt Orthonormalization Process Example

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The example of gram-schmidt orthonormalization process is as follows:

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Things to Remember

• Technically, the Gram–Schmidt orthonormalization process is a method of constructing an orthonormal basis from a set of vectors in an inner product space.
• Orthogonalization refers to the construction of a set of orthogonal vectors formed from a particular subspace.
• Orthonormalization is a type of orthogonalization that forms a unit vector.
• Before the term was coined, Pierre-Simon Laplace analyzed it.
• The Gram-Schmidt orthonormalization process can be stabilized using a technique called modified Gram-Schmidt or MGS.

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