# Periodic Functions

## Homework Statement

What kind of conditions do eigenvalues impose to ensure periodicity?
Is it plausible to say that irrational multiples of eigenvalues imply no harmonic oscillations, if so why?

## The Attempt at a Solution

Dick
Homework Helper
No, that doesn't make a whole lot of sense. Eigenvalues per se have nothing to do with periodicity.

$$\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -b & 0 \\ 0 & -b & 0 & -c \\ 0 & 0 & -c & 0\end{array}\right]$$

For any fixed a, I want to find b and c in terms of a such that lambda_i / lambda_ j is a rational number for every i,j=1,2,3,4 .

See, before I was working with

$$\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -a & 0 \\ 0 & -a & 0 & -a \\ 0 & 0 & -a & 0\end{array}\right]$$

But that gave me irrational e-values if I put it into Maple and I don't want that.

Last edited:
Dick
Homework Helper
$$\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -b & 0 \\ 0 & -b & 0 & -c \\ 0 & 0 & -c & 0\end{array}\right]$$

For any fixed a, I want to find b and c in terms of a such that lambda_i / lambda_ j is a rational number for every i,j=1,2,3,4 .

See, before I was working with

$$\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -a & 0 \\ 0 & -a & 0 & -a \\ 0 & 0 & -a & 0\end{array}\right]$$

But that gave me irrational e-values if I put it into Maple and I don't want that.

Now that makes more sense. It also looks like a hard question. How about a=1, b=0, c=1? Is that good enough? What's this for anyway?

Dick