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Difference between revisions of "Orthogonal transformation"

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A linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704001.png" /> of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer an orthonormal basis to an orthonormal one. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704002.png" /> is also a necessary and sufficient condition of orthogonality, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704003.png" /> is the conjugate and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704004.png" /> the inverse linear transformation.
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A linear transformation $A$ of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer an orthonormal basis to an orthonormal one. The equality $A^*=A^{-1}$ is also a necessary and sufficient condition of orthogonality, where $A^*$ is the conjugate and $A^{-1}$ the inverse linear transformation.
  
With respect to an orthonormal basis, orthogonal matrices correspond to orthogonal transformations and only to them. The eigen values of an orthogonal transformation are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704005.png" />, while the eigen vectors which correspond to different eigen values are orthogonal. The determinant of an orthogonal transformation is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704006.png" /> (special orthogonal transformation) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704007.png" /> (non-special orthogonal transformation). In the Euclidean plane, every special orthogonal transformation is a rotation, and its matrix in an appropriate orthonormal basis has the form
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With respect to an orthonormal basis, orthogonal matrices correspond to orthogonal transformations and only to them. The eigen values of an orthogonal transformation are equal to $\pm1$, while the eigen vectors which correspond to different eigen values are orthogonal. The determinant of an orthogonal transformation is equal to $+1$ (special orthogonal transformation) or $-1$ (non-special orthogonal transformation). In the Euclidean plane, every special orthogonal transformation is a rotation, and its matrix in an appropriate orthonormal basis has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704008.png" /></td> </tr></table>
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$$\begin{Vmatrix}\cos\phi&-\sin\phi\\\sin\phi&\hphantom{-}\cos\phi\end{Vmatrix},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o0704009.png" /> is the angle of the rotation; and every non-special orthogonal transformation is a reflection with respect to a straight line through the origin, and its matrix in an appropriate orthonormal basis has the form
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where $\phi$ is the angle of the rotation; and every non-special orthogonal transformation is a reflection with respect to a straight line through the origin, and its matrix in an appropriate orthonormal basis has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o07040010.png" /></td> </tr></table>
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$$\begin{Vmatrix}1&\hphantom{-}0\\0&-1\end{Vmatrix}.$$
  
In three-dimensional space, every special orthogonal transformation is a rotation around an axis, while every non-special orthogonal transformation is the product of such a rotation and a reflection in a perpendicular plane. In an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070400/o07040011.png" />-dimensional Euclidean space, orthogonal transformations also reduce to rotations and reflections (see [[Rotation|Rotation]]).
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In three-dimensional space, every special orthogonal transformation is a rotation around an axis, while every non-special orthogonal transformation is the product of such a rotation and a reflection in a perpendicular plane. In an arbitrary $n$-dimensional Euclidean space, orthogonal transformations also reduce to rotations and reflections (see [[Rotation|Rotation]]).
  
 
The set of all orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the [[Orthogonal group|orthogonal group]] of the given Euclidean space. The special orthogonal transformations form a normal subgroup in this group (the special orthogonal group).
 
The set of all orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the [[Orthogonal group|orthogonal group]] of the given Euclidean space. The special orthogonal transformations form a normal subgroup in this group (the special orthogonal group).

Latest revision as of 18:52, 18 September 2014

A linear transformation $A$ of a Euclidean space preserving the lengths (or, equivalently, the scalar product) of vectors. Orthogonal transformations and only they can transfer an orthonormal basis to an orthonormal one. The equality $A^*=A^{-1}$ is also a necessary and sufficient condition of orthogonality, where $A^*$ is the conjugate and $A^{-1}$ the inverse linear transformation.

With respect to an orthonormal basis, orthogonal matrices correspond to orthogonal transformations and only to them. The eigen values of an orthogonal transformation are equal to $\pm1$, while the eigen vectors which correspond to different eigen values are orthogonal. The determinant of an orthogonal transformation is equal to $+1$ (special orthogonal transformation) or $-1$ (non-special orthogonal transformation). In the Euclidean plane, every special orthogonal transformation is a rotation, and its matrix in an appropriate orthonormal basis has the form

$$\begin{Vmatrix}\cos\phi&-\sin\phi\\\sin\phi&\hphantom{-}\cos\phi\end{Vmatrix},$$

where $\phi$ is the angle of the rotation; and every non-special orthogonal transformation is a reflection with respect to a straight line through the origin, and its matrix in an appropriate orthonormal basis has the form

$$\begin{Vmatrix}1&\hphantom{-}0\\0&-1\end{Vmatrix}.$$

In three-dimensional space, every special orthogonal transformation is a rotation around an axis, while every non-special orthogonal transformation is the product of such a rotation and a reflection in a perpendicular plane. In an arbitrary $n$-dimensional Euclidean space, orthogonal transformations also reduce to rotations and reflections (see Rotation).

The set of all orthogonal transformations in a Euclidean space is a group with respect to multiplication of transformations — the orthogonal group of the given Euclidean space. The special orthogonal transformations form a normal subgroup in this group (the special orthogonal group).


Comments

See also Orthogonal matrix and Orthogonal group, and the references therein.

How to Cite This Entry:
Orthogonal transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_transformation&oldid=33318
This article was adapted from an original article by T.S. Pigolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article