Gram-Schmidt process

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In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rⁿ. Orthogonalization in this context means the following: we start with vectors v₁,...,v_k which are linearly independent and we want to find mutually orthogonal vectors u₁,...,u_k which generate the same subspace as the vectors v₁,...,v_k.

We denote the inner (dot) product of the two vectors u and v by (u . v). The Gram-Schmidt process works as follows:

u₁ = v₁

u₂ = v₂ - [(v₂ . u₁)/(u₁ . u₁)]u₁

u₃ = v₃ - [(v₃ . u₁)/(u₁ . u₁)]u₁ - [(v₃ . u₂)/(u₂ . u₂)]u₂

...

u_k = v_k - [(v_k . u₁)/(u₁ . u₁)]u₁ - [(v_k . u₂)/(u₂ . u₂)]u₂ - ... - [(v_k . u_k-1)/(u_k-1 . u_k-1)]u_k-1

To check that these formulas yield an orthogonal sequence, first compute (u₁ . u₂) by substituting the above formula for u₂: you will get zero. Then use this to compute (u₁ . u₃) again by substituting the formula for u₃: you will get zero. The general proof proceeds by mathematical induction.

Geometrically, this method proceeds as follows: to compute u_i, it projects v_i orthogonally onto the subspace U generated by u₁,...,u_i-1, which is the same as the subspace generated by v₁,...,v_i-1. u_i is then defined to be the difference between v_i and this projection, guaranteed to be orthogonal to all of the vectors in the subspace U.

If one is interested in an orthonormal system u₁,...,u_k (i.e. the vectors are mutually orthogonal and all have norm 1), then one can divide u_i by its norm (u_i . u_i).

The Gram-Schmidt process also applies to a linearly independent infinite sequence {v_i}_i. The result is an orthogonal (or orthonormal) sequence {u_i}'_i such that for natural number n: the algebraic span of v₁,...,v_n is the same as that of 'u₁,...,un''.

When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram-Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects.

The method is named for Jorgen Pedersen Gram and Erhard Schmidt, but is older, and to be found in the work of Laplace and Cauchy. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition.