# Matrix “Operators” and the Eigenvalue Problem Lab Using Maple

*label*Physics

*timer*Asked: Dec 5th, 2018

**Question description**

Physics 346—Laboratory

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

problems.

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.