Rotation matrix
First, one of the roots (or eigenvalues) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos (the negated linear term). This factorization is of interest for 33 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the “complex conjugates” are both 1, and for a 180 rotation they are both 1.) Furthermore, a similar factorization holds for any nn rotation matrix. If the dimension, n, is odd, there will be a “dangling” eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree n and thus n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n2 of them.
The sum of the entries on the main diagonal of a matrix is called the trace; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for 22 and 33 rotation matrices that the trace reveals the angle of rotation, , in the two-dimensional (sub-)space. For a 22 matrix the trace is 2 cos(), and for a 33 matrix it is 1+2 cos(). In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any 33 rotation matrix a rotation axis and an angle, and these completely determine the rotation.
Sequential angles
The constraints on a 22 rotation matrix imply that it must have the form
with a2+b2 = 1. Therefore we may set a = cos and b = sin , for some angle . To solve for it is not enough to look at a alone or b alone; we must consider both together to place the angle in the correct quadrant, using a two-argument arctangent function.
Now consider the first column of a 33 rotation matrix,
Although a2+b2 will probably not equal 1, but some value r2 < 1, we can use a slight variation of the previous computation to find a so-called Givens rotation that transforms the column to
zeroing b. This acts on the subspace spanned by the x and y axes. We can then repeat the process for the xz subspace to zero c. Acting on the full matrix, these two rotations produce the schematic form
Shifting attention to the second column, a Givens rotation of the yz subspace can now zero the z value. This brings the full matrix to the form
which is an identity matrix. Thus we have decomposed Q as
An nn rotation matrix will have (n1)+(n2)++2+1, or
entries below the diagonal to zero. We can zero them by extending the same idea of stepping through the columns with a series of rotations in a fixed sequence of planes. We conclude that the set of nn rotation matrices, each of which has n2 entries, can be parameterized by n(n1)/2 angles.
xzxw
xzyw
xyxw
xyzw
yxyw
yxzw
yzyw
yzxw
zyzw
zyxw
zxzw
zxyw
xzxb
yzxb
xyxb
zyxb
yxyb
zxyb
yzyb
xzyb
zyzb
xyzb
zxzb
yxzb
In three dimensions this restates in matrix form an observation made by Euler, so mathematicians call the ordered sequence of three angles Euler angles. However, the situation is somewhat more complicated than we have so far indicated. Despite the small dimension, we actually have considerable freedom in the sequence of axis pairs we use; and we also have some freedom in the choice of angles. Thus we find many different conventions employed when three-dimensional rotations are parameterized for physics, or medicine, or chemistry, or other disciplines. When we include the option of world axes or body axes, 24 different sequences are possible. And while some disciplines call any sequence Euler angles, others give different names (Euler, Cardano, Tait-Bryan, roll-pitch-yaw) to different sequences.
One reason for the large number of options is that, as noted previously, rotations in three dimensions (and higher) do not commute. If we reverse a given sequence of rotations, we get a different outcome. This also implies that we cannot compose two rotations by adding their corresponding angles. Thus Euler angles are not vectors, despite a similarity in appearance as a triple of numbers.
Nested dimensions
A 33 rotation matrix like
suggests a 22 rotation matrix,
is embedded in the upper left corner:
This is