Exponentiation of a matrix

[gentsagree]

I need to retrieve a finite rotation matrix (with cos and sin) from the exponentiation of the infinitesimal version of it.

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

My guess would be

$$\exp(\omega)=\sum_{k=0}\frac{\omega^{2k}}{2k!}+\sum_{k=0}\frac{\omega^{2k+1}}{(2k+1)!}$$

i.e. the even and odd contributions.

The notes I'm reading suggest instead:

$$\exp(\omega)=I+\sum_{k=1}\frac{\omega^{2k}}{2k!}+\sum_{k=1}\frac{\omega^{2k+1}}{(2k+1)!}$$

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?

 

[AlephZero]

I agree. Replacing the $\omega^0/0!$ term by $I$ is fair enough, but the second sum should be from $k = 0$.

It's probably just a typo.

 

[HallsofIvy]

?? The "linear term in $\omega$" is the term with $\omega^1$ which means it is the term in the second sum with k= 0: $\frac{\omega^{2(0)+ 1}}{(2(0)+ 1)!}= \omega$.

The linear approximation to $e^\omega$ is $I+ \omega$.

 

[FactChecker]

I agree. It's a typo. If you are doing this, don't forget that you can use the characteristic equation of the matrix ω to eliminate all the high powers.

 

[blue_raver22]

I would suggest that both approaches are valid and it ultimately depends on the specific application and context in which the finite rotation matrix is being used. Both approaches involve expanding the exponential function using a power series, but the difference lies in how the terms are organized and whether the identity matrix is included or not.

In the first approach, the identity matrix is not explicitly included in the series, but it is implicitly included in the first term of the even part. However, this may not be immediately clear and could potentially cause confusion when trying to retrieve the series expansion expressions for cos and sin.

In the second approach, the identity matrix is explicitly included as the first term, making it easier to see that it is part of the series. This could be helpful for understanding the overall structure of the series, but it may also make the series expression for cos and sin slightly more complicated.

Ultimately, I would recommend discussing with your colleagues or consulting with experts in the field to determine which approach would be most appropriate for your specific application. It's important to carefully consider the context and potential implications of each approach before making a decision.