Orthogonal Transformation (original) (raw)
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology
Alphabetical Index New in MathWorld
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,
 |
(1) |
|---|
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
satisfying the orthogonality condition
 |
(5) |
|---|
where Einstein summation has been used and 
is the Kronecker delta, is an orthogonal transformation. If 
is an orthogonal transformation, then 
.
See also
Improper Rotation, Inner Product, Lie Group, Linear Transformation, Lorentz Transformation,Matrix, Orthogonal Matrix,Orthogonal Group, Orthogonality Condition, Rotation
This entry contributed by Todd Rowland
Explore with Wolfram|Alpha
Cite this as:
Rowland, Todd. "Orthogonal Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/OrthogonalTransformation.html