I need to retrieve a finite rotation matrix (with cos and sin) from the exponentiation of the infinitesimal version of it.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose my infinitesimal matrix is ω. I then compute exp(ω).

My guess would be

[tex]\exp(\omega)=\sum_{k=0}\frac{\omega^{2k}}{2k!}+\sum_{k=0}\frac{\omega^{2k+1}}{(2k+1)!}[/tex]

i.e. the even and odd contributions.

The notes I'm reading suggest instead:

[tex]\exp(\omega)=I+\sum_{k=1}\frac{\omega^{2k}}{2k!}+\sum_{k=1}\frac{\omega^{2k+1}}{(2k+1)!}[/tex]

which looks weird to me; if I take the identity matrix I to be the k=0 contribution of the even part (ω^0=1), then I don't know where the term linear in ω is in the series any more. I think it's not there at all.

Even more: I do need the k=0 contributions later on to retrieve the series expansion expressions for cos and sin.

What do you think? Any comments?

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# Exponentiation of a matrix

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