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

`nθ`

.To calculate R

^{n}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 x

_{1}gives:

but

We could solve this:

Without writing it out, it looks like we just need to try a change of sign:

and λ

_{1}x

_{1}gives the same result:

Similarly, x

_{2}should be (i,-1) because R x

_{2}equals:

and

Everything checks.

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

`I`

:Let's also confirm that S Λ S

^{-1}= R. Part one is:

Step 2:

And it checks.

To do the exponentiation we just need to calculate Λ

^{n}. 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.

We're done!