Editing Orthogonal matrix (section)

===Lie algebra===
Suppose the entries of {{mvar|Q}} are differentiable functions of {{mvar|t}}, and that {{math|1=''t'' = 0}} gives {{math|1=''Q'' = ''I''}}. Differentiating the orthogonality condition
<math display="block">Q^\mathrm{T} Q = I </math>
yields
<math display="block">\dot{Q}^\mathrm{T} Q + Q^\mathrm{T} \dot{Q} = 0</math>

Evaluation at {{math|1=''t'' = 0}} ({{math|1=''Q'' = ''I''}}) then implies
<math display="block">\dot{Q}^\mathrm{T} = -\dot{Q} .</math>

In Lie group terms, this means that the [[Lie algebra]] of an orthogonal matrix group consists of [[skew-symmetric matrix|skew-symmetric matrices]]. Going the other direction, the [[matrix exponential]] of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).

For example, the three-dimensional object physics calls [[angular velocity]] is a differential rotation, thus a vector in the Lie algebra <math>\mathfrak{so}(3)</math> tangent to {{math|SO(3)}}. Given {{math|1='''ω''' = (''xθ'', ''yθ'', ''zθ'')}}, with {{math|1='''v''' = (''x'', ''y'', ''z'')}} being a unit vector, the correct skew-symmetric matrix form of {{mvar|'''ω'''}} is
<math display="block">
\Omega = \begin{bmatrix}
0 & -z\theta & y\theta \\
z\theta & 0 & -x\theta \\
-y\theta & x\theta & 0
\end{bmatrix} .</math>

The exponential of this is the orthogonal matrix for rotation around axis {{math|'''v'''}} by angle {{mvar|θ}}; setting {{math|1=''c'' = cos {{sfrac|''θ''|2}}}}, {{math|1=''s'' = sin {{sfrac|''θ''|2}}}},
<math display="block">\exp(\Omega) = \begin{bmatrix}
1  -  2s^2  +  2x^2 s^2  &  2xy s^2  -  2z sc  &  2xz s^2  +  2y sc\\
2xy s^2  +  2z sc  &  1  -  2s^2  +  2y^2 s^2  &  2yz s^2  -  2x sc\\
2xz s^2  -  2y sc  &  2yz s^2  +  2x sc  &  1  -  2s^2  +  2z^2 s^2   
\end{bmatrix}.</math>