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) |
(5) |
(6) |
(7) |
Let
and
be two orthogonal matrices. By the Orthogonality Condition, they satisfy
(8) |
(9) |
(10) |
The Eigenvalues of an orthogonal matrix must satisfy one of the following:
An orthogonal Matrix
is classified as proper
(corresponding to pure Rotation) if
(11) |
(12) |
See also Euler's Rotation Theorem, Orthogonal Transformation, Orthogonality Condition, Rotation, Rotation Matrix, Rotoinversion
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 Eric W. Weisstein