I’m trying to find the spectra (eigenvalues) of matrix
I start by generating its characteristic polynomial which yields:
Applying the Euler Formula:
At this point, I want to extract roots, the solutions for eigenvalues . One, from the LH term, is clearly . But that in the middle is stopping me from factorizing it via any way I can work out. I expect, since A was orthogonal, that the other two roots will be a complex conjugate pair but am not 100% on that – in any case, I can’t see how I can legally break that trig term into a complex one.
Thanks for the quick reply!
I get most of what you're saying and it looks good. However - one think I don't follow. You identify the discriminant as . However, when I try to determine it:
Now squaring gives , so I'm not sure how it converted to that sine function?
My guess is I got something wrong with my trig manipulations!
Just a few random thoughts: the eigenvalue problem has to do with operators whose action on an eigenvector is equivalent to multiplying the eigenvector by a scalar. Now, the matrix you've exhibited is a rotation about the z axis through an angle x. Therefore, I would expect any ol' vector along the z axis to be an eigenvector with eigenvalue 1. Any other vector could be an eigenvector with eigenvalue 1 if you rotated through radians, with . In addition, any vector in the xy plane could be an eigenvector with eigenvalue -1, if you rotate through an angle with . I don't think there are any other eigenvector/eigenvalue pairs. At least, none come to mind, because the fact is, that aside from a vector along the z axis, this rotation is going to change the direction of any other vector, unless with .
I just say all this by way of providing a double-check mechanism on your calculations, which seem to be in good hands.