It is also possible to use the trace of the rotation matrix.

That is, the elements of a rotation matrix are not all completely independent.

Third, it must also need to be shown that the above expression for is a rotation matrix.

In other words, they hold for any rotation matrix .

Analysis is often easier in terms of these generators, rather than the full rotation matrix.

The 4x4 rotation matrices have therefore 6 out of 16 independent components.

Using the same idea as before, we have a rotation matrix of:

This result can be obtained from a rotation matrix.

The interpretation of a rotation matrix can be subject to many ambiguities.

In that case the rotation matrix is time dependent.

The rotation matrix equivalent is given beneath each case.

In two dimensions every rotation matrix has the following form:

Generate a uniform angle and construct a 2x2 rotation matrix.

The product of two complex rotation matrices are given by:

The rotation matrix and the translation vector have three degrees of freedom each, in total six.

To be a proper rotation matrix it must also satisfy .

Bivectors are also related to the rotation matrix in n dimensions.

See Ambiguities in the definition of rotation matrices for more details.

We sometimes need to generate a uniformly distributed random rotation matrix.

This is similar to the rotation produced by the above mentioned 2-D rotation matrix.

Other rotation matrices can be obtained from these three using matrix multiplication.

They are related to the eigenvalues and eigenvectors of a rotation matrix.

Rotation matrices depend on the axis around which a point is to be rotated.

Where the left upper 3x3 matrix is the rotation matrix we just calculated.

It is also interesting to know that since is a rotation matrix: