Editing Gram–Schmidt process (section)

== Properties ==

Denote by <math> \operatorname{GS}(\mathbf{v}_1, \dots, \mathbf{v}_k) </math> the result of applying the Gram–Schmidt process to a collection of vectors <math> \mathbf{v}_1, \dots, \mathbf{v}_k </math>. This yields a map <math> \operatorname{GS} \colon (\R^n)^{k} \to (\R^n)^{k} </math>.

It has the following properties:

* It is continuous
* It is [[orientation (vector space)|orientation]] preserving in the sense that <math> \operatorname{or}(\mathbf{v}_1,\dots,\mathbf{v}_k) = \operatorname{or}(\operatorname{GS}(\mathbf{v}_1,\dots,\mathbf{v}_k)) </math>.
* It commutes with orthogonal maps:

Let <math> g \colon \R^n \to \R^n </math> be orthogonal (with respect to the given inner product). Then we have
<math display="block"> \operatorname{GS}(g(\mathbf{v}_1),\dots,g(\mathbf{v}_k)) = \left( g(\operatorname{GS}(\mathbf{v}_1,\dots,\mathbf{v}_k)_1),\dots,g(\operatorname{GS}(\mathbf{v}_1,\dots,\mathbf{v}_k)_k) \right) </math>

Further, a parametrized version of the Gram–Schmidt process yields a (strong) [[Retraction (topology)#Deformation retract and strong deformation retract|deformation retraction]] of the general linear group <math> \mathrm{GL}(\R^n)</math> onto the orthogonal group <math> O(\R^n)</math>.