Orthogonality Condition

A linear transformation

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

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

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

is said to be an Orthogonal Transformation if it satisfies the orthogonality condition

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

where Einstein Summation has been used and $\delta_{ij}$ is the Kronecker Delta.

See also Orthogonal Transformation

References

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