An orthogonal transformation is a linear transformation 
which preserves a symmetric inner product.
In particular, an orthogonal transformation (technically, an orthonormal transformation)
preserves lengths of vectors and angles between vectors,
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(1)
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In addition, an orthogonal transformation is either a rigid rotation or an improper rotation (a rotation followed by a flip). (Flipping and then rotating can be realized by first rotating in the reverse direction and then flipping.) Orthogonal transformations correspond to and may be represented using orthogonal matrices.
The set of orthonormal transformations forms the orthogonal group, and an orthonormal transformation can be realized by an orthogonal matrix.
Any linear transformation in three dimensions
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(2)
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(3)
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(4)
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satisfying the orthogonality condition
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(5)
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where Einstein summation has been used and 
is the Kronecker delta, is an orthogonal transformation.
If 
is an orthogonal transformation, then 
.
