Rotation Matrix - Properties of A Rotation Matrix

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.

Read more about this topic:  Rotation Matrix

Famous quotes containing the words properties of a, properties of, properties, rotation and/or matrix:

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The lazy manage to keep up with the earth’s rotation just as well as the industrious.”
    —Mason Cooley (b. 1927)

    ““The matrix is God?”
    “In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
    “If it has limits, it isn’t omnipotent.”
    “Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
    —William Gibson (b. 1948)