Periodic Functions Homework: Eigenvalues & Oscillations

• Nusc
In summary, the conversation discusses the conditions that eigenvalues impose for periodicity and the plausibility of irrational multiples of eigenvalues implying no harmonic oscillations. The speaker also mentions their attempt to find an equation for the eigenvalues and the difficulty of enforcing the condition that the ratio of roots is rational.

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

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

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?

You can certainly get an expression for the eigenvalues in terms of a,b and c. You get a quartic equation to solve, but it only has l^4, l^2 and a constant term. So you can solve it with the quadratic equation. But how you enforce the condition that the ratio of roots is rational, I have no idea.

1. What are eigenvalues and how are they related to periodic functions?

Eigenvalues are a mathematical concept used to describe the behavior of a linear transformation or an operator. They represent the values of the transformation or operator that do not change the direction of the corresponding eigenvectors. In the context of periodic functions, eigenvalues are used to describe the rate at which the function repeats itself, also known as the frequency.

2. How do I determine the eigenvalues of a periodic function?

To determine the eigenvalues of a periodic function, you will need to solve the characteristic equation of the function. This can be done by setting the function equal to its own derivative and solving for the values of x that satisfy the equation. These values of x will be the eigenvalues of the function.

3. What are oscillations and how do they relate to periodic functions?

Oscillations are a type of movement or fluctuation that occurs periodically around a central point. In the context of periodic functions, they refer to the repeated patterns or cycles that the function displays. The amplitude and frequency of these oscillations are determined by the eigenvalues of the function.

4. How can I use eigenvalues to solve problems involving periodic functions?

Eigenvalues can be used to determine the behavior and properties of periodic functions, such as their amplitude, frequency, and phase shift. They can also be used to solve differential equations involving periodic functions, as well as to analyze the stability of systems that exhibit periodic behavior.

5. Are there any real-life applications of periodic functions and eigenvalues?

Yes, periodic functions and eigenvalues have many real-life applications. They are used in fields such as physics, engineering, and economics to model and analyze natural phenomena, such as the motion of a pendulum, the oscillations of a spring, or the fluctuations in stock prices. They are also used in signal processing, image and sound compression, and data analysis.