The supplemental pages for this exercise can be found on D2L. They explain syntax for doing matrix
algebra on the computer so that you can perform rotational transformations and do eigenvalue
The eigenvalue problem is such an innocent little statement: Mu = μ u, and yet it has far-reaching
impact on many subfields in the physical sciences. Today you’ll study matrix operations and
then work out a coupled oscillator problem. The connection to the eigenvalue problem is not
immediately obvious, but is indeed there since the frequencies of the normal modes of oscillation
depend on the eigenvalues of a particular matrix.
First, convince yourself that two successive rotations about the z-axis through angles 1 and 2 is
equivalent to a single matrix operation (rotation) through angle ( 1 + 2). You’ll need to recall
Euler’s formula, which can be used to derive the useful expressions
sin ( 1 + 2) = sin 1cos 2 + cos 1sin 2
cos ( 1 + 2) = cos 1cos 2 – sin 1sin 2˙ .
This idea of successive transformations carries over to multiple operations which involves several
matrices multiplied together in the appropriate order. You’ll use this in what follows.