Orthogonal Transformation

Any linear transformation

$$x_1' = a_{11}x_1+a_{12}x_2+x_{13}x_3$$

$$x_2' = a_{21}x_1+a_{22}x_2+a_{23}x_3$$

$$x_3' = a_{31}x_1+a_{32}x_2+a_{33}x_3$$

satisfying the Orthogonality Condition

$$a_{ij}a_{ik} = \delta_{jk},$$

where Einstein Summation has been used and $\delta_{ij}$ is the Kronecker Delta, is called an orthogonal transformation.

Orthogonal transformations correspond to rigid Rotations (or Rotoinversions), and may be represented using Orthogonal Matrices. If $A:\Bbb{R}^n\to\Bbb{R}^n$ is an orthogonal transformation, then $\mathop{\rm det}(A)=\pm 1$.

See also Affine Transformation, Orthogonal Matrix, Orthogonality Condition, Rotation, Rotoinversion

References

Goldstein, H. ``Orthogonal Transformations.'' §4-2 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 132-137, 1980.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 104, 1993.