Editing Rotation matrix (section)

=== Skew parameters via Cayley's formula ===
{{main|Cayley transform|Skew-symmetric matrix}}
When an {{math|''n'' × ''n''}} rotation matrix {{mvar|Q}}, does not include a −1 eigenvalue, thus none of the planar rotations which it comprises are 180° rotations, then {{math|''Q'' + ''I''}} is an [[invertible matrix]]. Most rotation matrices fit this description, and for them it can be shown that {{math|(''Q'' − ''I'')(''Q'' + ''I'')<sup>−1</sup>}} is a [[skew-symmetric matrix]], {{mvar|A}}. Thus {{math|''A''<sup>T</sup> {{=}} −''A''}}; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, {{mvar|A}} contains {{math|{{sfrac|1|2}}''n''(''n'' − 1)}} independent numbers.

Conveniently, {{math|''I'' − ''A''}} is invertible whenever {{mvar|A}} is skew-symmetric; thus we can recover the original matrix using the ''[[Cayley transform]]'',
:<math> A \mapsto (I+A)(I-A)^{-1} , </math>
which maps any skew-symmetric matrix {{mvar|A}} to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180° rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions.

In three dimensions, for example, we have {{Harv|Cayley|1846}}
:<math>\begin{align}
    &\begin{bmatrix}
       0 & -z &  y \\
       z &  0 & -x \\
      -y &  x &  0
    \end{bmatrix} \mapsto \\[3pt]
  \quad \frac{1}{1 + x^2 + y^2 + z^2}
    &\begin{bmatrix}
      1 + x^2 - y^2 - z^2 & 2xy - 2z & 2y + 2xz \\
      2xy + 2z & 1 - x^2 + y^2 - z^2 & 2yz - 2x \\
      2xz - 2y & 2x + 2yz & 1 - x^2 - y^2 + z^2
    \end{bmatrix} .
\end{align}</math>

If we condense the skew entries into a vector, {{math|(''x'',''y'',''z'')}}, then we produce a 90° rotation around the {{mvar|x}}-axis for (1, 0, 0), around the {{mvar|y}}-axis for (0, 1, 0), and around the {{mvar|z}}-axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as {{math|''x'' → ∞}}, {{math|(''x'', 0, 0)}} does approach a 180° rotation around the {{mvar|x}} axis, and similarly for other directions.