Any Rotation can be given as a composition of rotations about three axes (Euler's Rotation Theorem), and thus
can be represented by a Matrix operating on a Vector,

(1) |

In a Rotation, a Vector must keep its original length, so it must be true that

(2) |

(3) |

(4) |

In order for this to hold, it must be true that

(5) |

(6) |

(7) |

Let
and
be two orthogonal matrices. By the Orthogonality Condition, they satisfy

(8) |

(9) |

(10) |

so the product of two orthogonal matrices is also orthogonal.

The Eigenvalues of an orthogonal matrix must satisfy one of the following:

- 1. All Eigenvalues are 1.
- 2. One Eigenvalue is 1 and the other two are .
- 3. One Eigenvalue is 1 and the other two are Complex Conjugates of the form and .

An orthogonal Matrix
is classified as proper
(corresponding to pure Rotation) if

(11) |

(12) |

**References**

Arfken, G. ``Orthogonal Matrices.'' *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 191-205, 1985.

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

© 1996-9

1999-05-26