Orthogonal Transformation

An orthogonal transformation is a linear transformation T:V->V which preserves a symmetric inner product. In particular, an orthogonal transformation (technically, an orthonormal transformation) preserves lengths of vectors and angles between vectors,


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


where Einstein summation has been used and delta_(ij) is the Kronecker delta, is an orthogonal transformation. If A:R^n->R^n is an orthogonal transformation, then det(A)=+/-1.

See also

Improper Rotation, Inner Product, Lie Group, Linear Transformation, Lorentz Transformation, Matrix, Orthogonal Matrix, Orthogonal Group, Orthogonality Condition, Spin Group, Rotation, Symmetric Quadratic Form

This entry contributed by Todd Rowland

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Rowland, Todd. "Orthogonal Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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