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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.

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

<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>

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

<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>

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]]).

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).



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See also [[Orthogonal matrix|Orthogonal matrix]] and [[Orthogonal group|Orthogonal group]], and the references therein.