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## 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?

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- Thread starter Nusc
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- #1

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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?

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Dick

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

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[tex]

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

[/tex]

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

[tex]

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

[/tex]

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

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

[/tex]

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

[tex]

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

[/tex]

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

Last edited:

- #4

Dick

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[tex]

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

[/tex]

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

[tex]

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

[/tex]

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?

- #5

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But forget that for now, how would I show that the eigenvalues are irrational for n>3 ? That's why I asked this guy in this thread https://www.physicsforums.com/showthread.php?t=224954&page=3

how to find an equation for the eigenvalues.

- #6

Dick

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