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,
In a Rotation, a Vector must keep its original length, so it must be true that
be two orthogonal matrices. By the Orthogonality Condition, they satisfy
The Eigenvalues of an orthogonal matrix must satisfy one of the following:
An orthogonal Matrix
is classified as proper
(corresponding to pure Rotation) if
See also Euler's Rotation Theorem, Orthogonal Transformation, Orthogonality Condition, Rotation, Rotation Matrix, Rotoinversion
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.
Goldstein, H. ``Orthogonal Transformations.'' §4-2 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 132-137, 1980.
© 1996-9 Eric W. Weisstein