In Strang section 6.2 (first challenge problem), we are asked to prove the reasonable supposition:
That is, n sequential applications of the R vector should give the same result as a single rotation through an angle of
To calculate Rn we'll need the eigenvalues and eigenvectors of R. The interest of this problem comes from consideration of the following question:
what sort of vector could have the same direction after rotation by an arbitrary angle θ?
Use the standard formula:
We need the quadratic equation to find λ:
Strang gives the eigenvectors as
However, I think this is a misprint since R x1 gives:
We could solve this:
Without writing it out, it looks like we just need to try a change of sign:
and λ1 x1 gives the same result:
Similarly, x2 should be (i,-1) because R x2 equals:
Now that we have the eigenvalues and eigenvectors, we need to diagonalize R. We construct the columns of S directly from the eiegenvectors:
Λ is just
To get S-1 let's try the trick for 2 x 2's:
and multipy to confirm we do get
Let's also confirm that S Λ S-1 = R. Part one is:
And it checks.
To do the exponentiation we just need to calculate Λn. So what is:
Strang drops another hint by reminding us of Euler's formula:
which is the whole point! But what about:
If we compare -θ with θ then:
Now all we have to do is convert back to sine and cosine:
The big computation we did above (to check that S Λ S-1 = R, is exactly the same with the substitution of nθ for θ.