Gram–Schmidt process

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.

The method is named for Jørgen 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.

The Gram-Schmidt process is numerically unstable: when implemented on a computer, the vectors u_k are not quite orthogonal because of rounding errors, and for the Gram-Schmidt process the loss of orthogonality is particularly bad. Therefore one usually prefers to use Householder transformations or Givens rotations.

The Gram-Schmidt process

We define the projection operator by

$\mathrm {proj} _{\mathbf {v} }\,\mathbf {u} ={\langle \mathbf {v} ,\mathbf {u} \rangle \over \langle \mathbf {v} ,\mathbf {v} \rangle }\mathbf {v} .$

The Gram-Schmidt process then works as follows:

	$\mathbf {u} _{1}=\mathbf {v} _{1},$		$\mathbf {e} _{1}={\mathbf {u} _{1} \over ||\mathbf {u} _{1}||}$
	$\mathbf {u} _{2}=\mathbf {v} _{2}-\mathrm {proj} _{\mathbf {e} _{1}}\,\mathbf {v} _{2},$		$\mathbf {e} _{2}={\mathbf {u} _{2} \over ||\mathbf {u} _{2}||}$
	$\mathbf {u} _{3}=\mathbf {v} _{3}-\mathrm {proj} _{\mathbf {e} _{1}}\,\mathbf {v} _{3}-\mathrm {proj} _{\mathbf {e} _{2}}\,\mathbf {v} _{3},$		$\mathbf {e} _{3}={\mathbf {u} _{3} \over ||\mathbf {u} _{3}||}$
	$\vdots$		$\vdots$
	$\mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\mathrm {proj} _{\mathbf {e} _{j}}\,\mathbf {v} _{k},$		$\mathbf {e} _{k}={\mathbf {u} _{k} \over ||\mathbf {u} _{k}||}$

The sequence u₁,...,u_k is the required system of orthogonal vectors, and the normalized vectors e₁,...,e_k form an orthonormal system.

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.

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,...,u_n.

Example

Consider the following set of vectors in Rⁿ (with the conventional inner product)

$S=\lbrace \mathbf {v} _{1}={\begin{pmatrix}3\\1\end{pmatrix}},\mathbf {v} _{2}={\begin{pmatrix}2\\2\end{pmatrix}}\rbrace .$

Now, perform Gram-Schmidt, to obtain an orthogonal set of vectors:

$\mathbf {u} _{1}=\mathbf {v} _{1},\qquad \mathbf {e} _{1}=1/{\sqrt {10}}{\begin{pmatrix}3,1\end{pmatrix}}$

$\mathbf {u} _{2}=\mathbf {v} _{2}-\mathrm {proj} _{\mathbf {e} _{1}}\,\mathbf {v} _{2}={\begin{pmatrix}2\\2\end{pmatrix}}-\mathrm {proj} _{1/{\sqrt {10}}{\begin{pmatrix}3,1\end{pmatrix}}}\,{\begin{pmatrix}2\\2\end{pmatrix}}={\begin{pmatrix}-2/5,6/5\end{pmatrix}}$

We check that the vectors u₁ and u₂ are indeed orthogonal:

$\langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{pmatrix}3\\1\end{pmatrix}},{\begin{pmatrix}-2/5\\6/5\end{pmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0.$