Consider the following problem, which occurs frequently in both mathematics and physics:
Problems of this form are referred to as eigenvalue problems. Eigen is the
German word for "specific", so literally translated eigenvalue means
"specific value".
Usually, A in these problems is a matrix, however, it can also be an
operator (a function that acts on a function, e.g. a partial derivative
with respect to time). The way to read this problem is "For what values
of the vector x will the operator A applied to x return a multiple of x,
and what are those multiples?" The values of the vector x are the
eigenvectors of the problem, and the multiples are the eigenvalues.
They are the specific vectors and values that make the statement true.
Exercise
Consider the following problem: An artist is commissioned to design a
public sculpture with a theme of pendulum motion. It is decided that
the sculpture will feature four pendulums of different length swinging
in motion, such that the longest pendulum takes 10 seconds to complete
a swing, and that the other pendulums will in that time take 2, 3, and
4 swings respectively, but all will return to their original position
every 10 seconds.
What are the eigenvalues that will satisfy the
artist's design?
Before constructing the sculpture, the artist
becomes concerned that the small angle approximation might not be
valid in this case. How would relaxing the small angle approximation
change your solution?
Use the numerical model of
pendulum motion
to solve this problem.