Rotation matrix
above discussion can be generalised to any number of dimensions. For any rotation matrix and I, the identity in
Examples
The 22 rotation matrix
corresponds to a 90 planar rotation.
The transpose of the 22 matrix
is its inverse, but since its determinant is 1 this is not a rotation matrix; it is a reflection across the line 11y = 2x.
The 33 rotation matrix
corresponds to a 30 rotation around the x axis in three-dimensional space.
The 33 rotation matrix
corresponds to a rotation of approximately 74 around the axis (13,23,23) in three-dimensional space.
The 33 permutation matrix
is a rotation matrix, as is the matrix of any even permutation, and rotates through 120 about the axis x = y = z.
The 33 matrix
has determinant +1, but its transpose is not its inverse, so it is not a rotation matrix.
The 43 matrix
is not square, and so cannot be a rotation matrix; yet MTM yields a 33 identity matrix (the columns are orthonormal).
The 44 matrix
describes an isoclinic rotation, a rotation through equal angles (180) through two orthogonal planes.
The 55 rotation matrix
rotates vectors in the plane of the first two coordinate axes 90, rotates vectors in the plane of the next two axes 180, and leaves the last coordinate axis unmoved.
Geometry
In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave “handedness” unchanged. By contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, and a glide reflection does both.
A rotation that does not leave “handedness” unchanged is called an Improper Rotation or a Rotoinversion
If we take the fixed point as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement from the origin. Thus we may work with the vector space of displacements instead of the points themselves. Now suppose (p1,,pn) are the coordinates of the vector p from the origin, O, to point P. Choose an orthonormal basis for our coordinates; then the squared distance to P, by Pythagoras, is
which we can compute using the matrix multiplication
A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties we can show that a rotation is a linear transformation of the vectors, and thus can be written in matrix form, Qp. The fact that a rotation preserves, not just ratios, but distances themselves, we can state as
or
Because this equation holds for all vectors, p, we conclude that every rotation matrix, Q, satisfies the orthogonality condition,
Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,
Equally important, we can show that any matrix satisfying these two conditions acts as a rotation.
Multiplication
The inverse of a rotation matrix is its transpose, which is also a rotation matrix:
The product of two rotation matrices is a rotation matrix:
For n greater than 2, multiplication of nn rotation matrices is not commutative.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the nn rotation matrices form a group, which for n > 2 is non-abelian. Called a special orthogonal group, and denoted by SO(n), SO(n,R), SOn, or SOn(R), the group of nn rotation matrices is isomorphic to the group of rotations in an n-dimensional space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.
Ambiguities
Alias and alibi rotations
The interpretation of a rotation matrix can be subject to many ambiguities.
Alias or alibi transformation
The change in a vector’s coordinates can indicate a turn of the coordinate system (alias) or a turn of the vector (alibi).
Right- or left-handed coordinates
The matrix can be with respect to a right-handed or left-handed coordinate system.
World or body axes
The coordinate axes can be fixed or rotate with a body.
Vectors or forms
The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.
In most cases the effect of the ambiguity is to transpose or invert the matrix.
Decompositions
Independent planes
Consider the 33 rotation matrix
If Q acts in a certain direction, v, purely as a scaling by a factor , then we have
so that
Thus is a root of the characteristic polynomial for Q,
Two features are noteworthy.