I have a matrix equation, taken from Wikipedia (Infinitesimale Drehungen), that looks not that complicated (note $a$ is a scalar, actually an angle as input paramter for the rotation matrix):
$$ R(a)=\exp(aJ) $$
In my case I would like to obtain $J$ while $R(a)$ is given: $$ R(a)=\left( \begin{array}{ccc} \cos\left(\sqrt{5}a\right)&-\frac{2\sin\left(\sqrt{5}a\right)}{\sqrt{5}}&\frac{\sin\left(\sqrt{5}a\right)}{\sqrt{5}}\\ \frac{2\sin\left(\sqrt{5}a\right)}{\sqrt{5}} & \frac{1}{5}\left(4\cos\left(\sqrt{5}a\right)+1\right) & -\frac{2}{5}\left(\cos\left(\sqrt{5}a\right)-1\right)\\ -\frac{\sin\left(\sqrt{5}a\right)}{\sqrt{5}} & -\frac{2}{5}\left(\cos\left(\sqrt{5}a\right)-1\right) & \frac{1}{5}\left(\cos\left(\sqrt{5}a\right)+4\right) \\ \end{array} \right) $$
I tried the following J[a_] := MatrixLog[R[a]]/a
which does not work. Based on the equation $J=\left.\frac{dR(a)}{da}\right|_{a=0}$ that is also provided on the above-given Wikipedia page, I tried J[a_] := D[R[a], a]
as well, which did not worked too.
My full listing looks as follows:
ClearAll["Global`*"];
R[a_] := {{Cos[\[Sqrt]5 a], -((2 Sin[\[Sqrt]5 a])/(\[Sqrt]5)), Sin[\[Sqrt]5 a]/(\[Sqrt]5)},
{(2 Sin[\[Sqrt]5 a])/(\[Sqrt]5), (1/5) (1 + 4 Cos[\[Sqrt]5 a]), -(2/5) (-1 + Cos[\[Sqrt]5 a])},
{-(Sin[\[Sqrt]5 a]/(\[Sqrt]5)), -(2/5) (-1 + Cos[\[Sqrt]5 a]), (1/5) (4 + Cos[\[Sqrt]5 a])}};
J[a_] := MatrixLog[R[a]]/a;
J[a_] := D[R[a], a];
FullSimplify[MatrixExp[a*J]]
FullSimplify[Limit[MatrixPower[IdentityMatrix[3] + (a/n) J, n], n -> Infinity]]
The last two Print
statements should yield the original matrix $R(a)$.
I would be greatful for any help on obtaining the Matrix $J$.
Asymptotic[R[a] - MatrixExp[a R'[a]] // Simplify, a -> 0]
shows that expression is smallO[a^2]
$\endgroup$Print
is unnecessary in expressions likePrint[FullSimplify[MatrixExp[a*J]]];
Just don't suppress the output with the semi-colon. $\endgroup$