Properties of A Rotation Matrix
In three dimensions, for any rotation matrix, where a is a rotation axis and θ a rotation angle,
-
- (i.e., is an orthogonal matrix)
- (i.e, the determinant of is 1)
- (where is the identity matrix)
- The eigenvalues of are
-
- where i is the standard imaginary unit with the property
- The trace of is equivalent to the sum of its eigenvalues.
Some of these properties can be generalised to any number of dimensions. In other words, they hold for any rotation matrix .
For instance, in two dimensions the properties hold with the following exceptions:
-
- a is not a given axis, but a point (rotation center) which must coincide with the origin of the coordinate system in which the rotation is represented.
- Consequently, the four elements of the rotation matrix depend only on θ, hence we write, rather than
- The eigenvalues of are
- The trace of is equivalent to the sum of its eigenvalues.
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—John Locke (1632–1704)
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