Editing Rotation matrix (section)

=== Sequential angles ===
The constraints on a {{nowrap|2 × 2}} rotation matrix imply that it must have the form
:<math>Q = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}</math>
with {{math|''a''<sup>2</sup> + ''b''<sup>2</sup> {{=}} 1}}. Therefore, we may set {{math|''a'' {{=}} cos ''θ''}} and {{math|''b'' {{=}} sin ''θ''}}, for some angle {{mvar|θ}}. To solve for {{mvar|θ}} it is not enough to look at {{mvar|a}} alone or {{mvar|b}} alone; we must consider both together to place the angle in the correct [[Cartesian coordinate system#Cartesian coordinates in two dimensions|quadrant]], using a [[atan2|two-argument arctangent]] function.

Now consider the first column of a {{nowrap|3 × 3}} rotation matrix,
:<math>\begin{bmatrix}a\\b\\c\end{bmatrix} . </math>
Although {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}} will probably not equal 1, but some value {{math|''r''<sup>2</sup> < 1}}, we can use a slight variation of the previous computation to find a so-called [[Givens rotation]] that transforms the column to
:<math>\begin{bmatrix}r\\0\\c\end{bmatrix} , </math>
zeroing {{mvar|b}}. This acts on the subspace spanned by the {{mvar|x}}- and {{mvar|y}}-axes. We can then repeat the process for the {{mvar|xz}}-subspace to zero {{mvar|c}}. Acting on the full matrix, these two rotations produce the schematic form
:<math>Q_{xz}Q_{xy}Q = \begin{bmatrix}1&0&0\\0&\ast&\ast\\0&\ast&\ast\end{bmatrix} . </math>
Shifting attention to the second column, a Givens rotation of the {{mvar|yz}}-subspace can now zero the {{mvar|z}} value. This brings the full matrix to the form
:<math>Q_{yz}Q_{xz}Q_{xy}Q = \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} , </math>
which is an identity matrix. Thus we have decomposed {{mvar|Q}} as
:<math>Q = Q_{xy}^{-1}Q_{xz}^{-1}Q_{yz}^{-1} . </math>

An {{math|''n'' × ''n''}} rotation matrix will have {{math|(''n'' − 1) + (''n'' − 2) + ⋯ + 2 + 1}}, or
:<math>\sum_{k=1}^{n-1} k = \frac{1}{2}n(n - 1) </math>
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 {{math|''n'' × ''n''}} rotation matrices, each of which has {{math|''n''<sup>2</sup>}} entries, can be parameterized by {{math|{{sfrac|1|2}}''n''(''n'' − 1)}} angles.

{| border="1" cellspacing="0" cellpadding="4" style="float:right; margin-left:1em"
|-
| {{math|''xzx''<sub>w</sub>}} || {{math|''xzy''<sub>w</sub>}} || {{math|''xyx''<sub>w</sub>}} || {{math|''xyz''<sub>w</sub>}}
|-
| {{math|''yxy''<sub>w</sub>}} || {{math|''yxz''<sub>w</sub>}} || {{math|''yzy''<sub>w</sub>}} || {{math|''yzx''<sub>w</sub>}}
|-
| {{math|''zyz''<sub>w</sub>}} || {{math|''zyx''<sub>w</sub>}} || {{math|''zxz''<sub>w</sub>}} || {{math|''zxy''<sub>w</sub>}}
|-
| {{math|''xzx''<sub>b</sub>}} || {{math|''yzx''<sub>b</sub>}} || {{math|''xyx''<sub>b</sub>}} || {{math|''zyx''<sub>b</sub>}}
|-
| {{math|''yxy''<sub>b</sub>}} || {{math|''zxy''<sub>b</sub>}} || {{math|''yzy''<sub>b</sub>}} || {{math|''xzy''<sub>b</sub>}}
|-
| {{math|''zyz''<sub>b</sub>}} || {{math|''xyz''<sub>b</sub>}} || {{math|''zxz''<sub>b</sub>}} || {{math|''yxz''<sub>b</sub>}}
|}
In three dimensions this restates in matrix form an observation made by [[Leonhard Euler|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 (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 [[vector space|vectors]], despite a similarity in appearance as a triplet of numbers.