Why, why, why, why …?

I bet you know this situation. You encounter an interesting theorem, but you are not 100% sure why it is true. You find a proof for this theorem only to see it refers to another theorem which is also interesting, but you are also not 100% sure why is it true. You can go on and on with this exploring seemingly unrelated things and admiring how everything is connected with everything :). I personally love when I do this stuff because I think it gives me a lot of insight into things.

I can give an example of this which I have really enjoyed a few months ago. I have asked myself this question: Why can every possible rotation in 3D space be expressed in axis-angle representation?

When you look at this statement it seems reasonable to be true, but there is always a seed of doubt so I have started looking for a proof. The proof I have found was really elegant. It stated that because every rotation in 3D space can be expressed by a 3×3 matrix and this matrix has to have at least  one real eigenvalue, you can pick one of the eigenvectors of the matrix as the direction vector of the axis of rotation.

I have really liked this proof but I had to ask the next question myself: Why there has to be at least one real eigenvalue? The answer came a few seconds after: Because the characteristic polynomial is of order 3 and these polynomial have at least one real root. And because I was already in this mood I have spent the next few minutes experimenting with odd polynomials to really prove that there has to be at least one real root. This is what a simple question about rotations has led me to that day.




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