Editing Rotation matrix (section)

=== Nested dimensions ===
A {{nowrap|3 × 3}} rotation matrix such as
:<math>Q_{3 \times 3} = \begin{bmatrix}
   \cos \theta & -\sin \theta & {\color{CadetBlue}0} \\
   \sin \theta &  \cos \theta & {\color{CadetBlue}0} \\
  {\color{CadetBlue}0} & {\color{CadetBlue}0} & {\color{CadetBlue}1}
\end{bmatrix}</math>

suggests a {{nowrap|2 × 2}} rotation matrix,
:<math>Q_{2 \times 2} =
  \begin{bmatrix}
    \cos \theta & -\sin \theta \\ 
    \sin \theta & \cos \theta
  \end{bmatrix},
</math>

is embedded in the upper left corner:
:<math>Q_{3 \times 3} = \left[ \begin{matrix} Q_{2 \times 2} & \mathbf{0} \\ \mathbf{0}^\mathsf{T} & 1 \end{matrix} \right].</math>

This is no illusion; not just one, but many, copies of {{mvar|n}}-dimensional rotations are found within {{math|(''n'' + 1)}}-dimensional rotations, as [[subgroup]]s. Each embedding leaves one direction fixed, which in the case of {{nowrap|3 × 3}} matrices is the rotation axis. For example, we have
:<math>\begin{align}
  Q_{\mathbf{x}}(\theta) &= \begin{bmatrix}
      {\color{CadetBlue}1} & {\color{CadetBlue}0} & {\color{CadetBlue}0} \\
      {\color{CadetBlue}0} &  \cos \theta & -\sin \theta \\
      {\color{CadetBlue}0} &  \sin \theta & \cos \theta
    \end{bmatrix}, \\[8px]
  Q_{\mathbf{y}}(\theta) &= \begin{bmatrix}
       \cos \theta & {\color{CadetBlue}0} & \sin \theta \\
       {\color{CadetBlue}0} & {\color{CadetBlue}1} & {\color{CadetBlue}0} \\
       -\sin \theta & {\color{CadetBlue}0} & \cos \theta
    \end{bmatrix}, \\[8px]
  Q_{\mathbf{z}}(\theta) &= \begin{bmatrix}
       \cos \theta & -\sin \theta & {\color{CadetBlue}0} \\
       \sin \theta & \cos \theta & {\color{CadetBlue}0} \\
       {\color{CadetBlue}0} & {\color{CadetBlue}0} & {\color{CadetBlue}1}
    \end{bmatrix},
\end{align}</math>

fixing the {{mvar|x}}-axis, the {{mvar|y}}-axis, and the {{mvar|z}}-axis, respectively. The rotation axis need not be a coordinate axis; if {{math|'''u''' {{=}} (''x'',''y'',''z'')}} is a unit vector in the desired direction, then
:<math>\begin{align}
     Q_\mathbf{u}(\theta)
  &= \begin{bmatrix}
        0 & -z &  y \\
        z &  0 & -x \\
       -y &  x &  0
     \end{bmatrix} \sin\theta + \left(I - \mathbf{u}\mathbf{u}^\mathsf{T}\right) \cos\theta + \mathbf{u}\mathbf{u}^\mathsf{T} \\[8px]
  &= \begin{bmatrix}
        \left(1 - x^2\right) c_\theta + x^2 &
           -z s_\theta - x y c_\theta + x y &
            y s_\theta - x z c_\theta + x z \\
            z s_\theta - x y c_\theta + x y &
        \left(1 - y^2\right) c_\theta + y^2 &
           -x s_\theta - y z c_\theta + y z \\
           -y s_\theta - x z c_\theta + x z &
            x s_\theta - y z c_\theta + y z &
        \left(1 - z^2\right) c_\theta + z^2
     \end{bmatrix} \\[8px]
  &= \begin{bmatrix}
       x^2 (1 - c_\theta) +   c_\theta &
       x y (1 - c_\theta) - z s_\theta &
       x z (1 - c_\theta) + y s_\theta \\
       x y (1 - c_\theta) + z s_\theta &
       y^2 (1 - c_\theta) +   c_\theta &
       y z (1 - c_\theta) - x s_\theta \\
       x z (1 - c_\theta) - y s_\theta &
       y z (1 - c_\theta) + x s_\theta &
       z^2 (1 - c_\theta) + c_\theta
     \end{bmatrix}, 
\end{align}</math>

where {{math|''c<sub>θ</sub>'' {{=}} cos ''θ''}}, {{math|''s<sub>θ</sub>'' {{=}} sin ''θ''}}, is a rotation by angle {{mvar|θ}} leaving axis {{math|'''u'''}} fixed.

A direction in {{math|(''n'' + 1)}}-dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, {{math|''S''<sup>''n''</sup>}}. Thus it is natural to describe the rotation group {{math|SO(''n'' + 1)}} as combining {{math|SO(''n'')}} and {{math|''S''<sup>''n''</sup>}}. A suitable formalism is the [[fiber bundle]],
:<math>SO(n) \hookrightarrow SO(n + 1) \to S^n ,</math>

where for every direction in the base space, {{math|''S''<sup>''n''</sup>}}, the fiber over it in the total space, {{math|SO(''n'' + 1)}}, is a copy of the fiber space, {{math|SO(''n'')}}, namely the rotations that keep that direction fixed.

Thus we can build an {{math|''n'' × ''n''}} rotation matrix by starting with a {{nowrap|2 × 2}} matrix, aiming its fixed axis on {{math|''S''<sup>2</sup>}} (the ordinary sphere in three-dimensional space), aiming the resulting rotation on {{math|''S''<sup>3</sup>}}, and so on up through {{math|''S''<sup>''n''−1</sup>}}. A point on {{math|''S''<sup>''n''</sup>}} can be selected using {{mvar|n}} numbers, so we again have {{math|{{sfrac|1|2}}''n''(''n'' − 1)}} numbers to describe any {{math|''n'' × ''n''}} rotation matrix.

In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of {{math|''n'' − 1}} Givens rotations brings the first column (and row) to {{nowrap|(1, 0, ..., 0)}}, so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave {{nowrap|(1, 0, ..., 0)}} fixed.