Gram-Schmidt process

From Academic Kids

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,...,vk which are linearly independent and we want to find mutually orthogonal vectors u1,...,uk which generate the same subspace as the vectors v1,...,vk.

The method is named for Jrgen 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 uk 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

<math>\mathrm{proj}_{\mathbf{v}}\,\mathbf{u} = {\langle \mathbf{v}, \mathbf{u}\rangle\over\langle \mathbf{v}, \mathbf{v}\rangle}\mathbf{v}. <math>

The Gram-Schmidt process then works as follows:

<math>\mathbf{u}_1 = \mathbf{v}_1,<math> <math>\mathbf{e}_1 = {\mathbf{u}_1 \over ||\mathbf{u}_1||}<math>
<math>\mathbf{u}_2 = \mathbf{v}_2-\mathrm{proj}_{\mathbf{e}_1}\,\mathbf{v}_2, <math> <math>\mathbf{e}_2 = {\mathbf{u}_2 \over ||\mathbf{u}_2||}<math>
<math>\mathbf{u}_3 = \mathbf{v}_3-\mathrm{proj}_{\mathbf{e}_1}\,\mathbf{v}_3-\mathrm{proj}_{\mathbf{e}_2}\,\mathbf{v}_3, <math> <math>\mathbf{e}_3 = {\mathbf{u}_3 \over ||\mathbf{u}_3||}<math>
<math>\vdots<math> <math>\vdots<math>
<math>\mathbf{u}_k = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{e}_j}\,\mathbf{v}_k, <math> <math>\mathbf{e}_k = {\mathbf{u}_k\over||\mathbf{u}_k||}<math>

The sequence u1,...,uk is the required system of orthogonal vectors, and the normalized vectors e1,...,ek 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 ui, it projects vi orthogonally onto the subspace U generated by u1,...,ui-1, which is the same as the subspace generated by v1,...,vi-1. ui is then defined to be the difference between vi 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 {vi}i. The result is an orthogonal (or orthonormal) sequence {ui}i such that for natural number n: the algebraic span of v1,...,vn is the same as that of u1,...,un.

Example

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

<math>S = \lbrace\mathbf{v}_1=\begin{pmatrix} 3 \\ 1\end{pmatrix}, \mathbf{v}_2=\begin{pmatrix}2 \\2\end{pmatrix}\rbrace.<math>

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

<math>\mathbf{u}_1=\mathbf{v}_1,\qquad\mathbf{e}_1=1/\sqrt{10}\begin{pmatrix}3, 1\end{pmatrix}<math>
<math>\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}<math>

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

<math>\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 = -\frac65 + \frac65 = 0.<math>de:Gram-Schmidtsches Orthogonalisierungsverfahren

it:Ortogonalizzazione di Gram - Schmidt ja:グラム・シュミットの正規直交化法

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