Properties of the Rotation Matrix

The rotation matrix will often be denoted in the text, to make its notation more compact, as a C language array:

$$\mathbf{R} = \begin{bmatrix} r_{0} & r_{1} & r_{2} \\ r_{3} & r_{4} & r_{5} \\ r_{6} & r_{7} & r_{8} \end{bmatrix}$$

The rotation matrix is highly overparameterized: its 9 linearly independent parameters are actually generated nonlinearly from 3 variables (see the appendix).

Without explicitly specifying the angles from which the matrix is generated, it is still possible to impose some additional constraints. The rotation matrix preserves distances because it is orthonormal and $\det(\mathbf{R})=1$. Every row and every column must have unit norm, and the rows and columns are mutually orthonormal, since they form orthonormal bases of the space. Therefore, given two row or column vectors of matrix $\mathbf{r}_1$, $\mathbf{r}_2$, it is possible to determine the third basis vector as the cross product of the other two:

$$\mathbf{r}_3 = \mathbf{r}_1 \times \mathbf{r}_2$$ (9.29)

Likewise, the dot product between any two row vectors or any two column vectors must be zero, since they are orthogonal to each other.