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Welcome back after this practice section where we have found out that the eigenvalues of the Matrix

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eight are given by these expressions here.

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So you see here on the numerical values, but there's point for one, for two, this is clearly a square

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root of two.

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And you could also figure this out by just calculating analytically the roots of this polynomial here,

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which I have done, and the analytical solutions are indeed two plus get rid of two, two and two minus

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square root of two.

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And we've learned before that the eigen frequencies omega aren't given by the square root of K over

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m times lambda and in our case, Keenum are one.

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So we just have to calculate here to square roots of these values.

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OK.

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So these are the eigen frequencies, and we need to understand now what these IGen frequencies mean.

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And one thing that we can do before is we can analyze the EIGEN system.

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So maybe you have heard from this by mathematics.

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That's a matrix does not only have eigenvalues, but it also has eigenvectors which are closely related

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to the transformation that we have talked about before.

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And to calculate the system, we have to write and p dot pulling out.

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Off our matrix and then dot h.

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So not Ike, but just Ike.

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And this gives us the whole eigen system where these ones here are the eigenvalues.

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So the same as before.

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And these ones here are the corresponding eigenvectors.

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So this is here's what you get analytically.

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These are the three vectors and you can see they correspond to these components that you can see here.

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One over square root of two is approximately zero point seven.

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Oh, seven.

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OK, so these basically correspond one to one to these frequencies.

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So the first vector corresponds to the largest frequency, which would be this one here.

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And the values here day describe how the individual oscillators move with respect to each other.

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So a preferred mode of oscillation would be each one of those three.

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But depending on the frequency, the oscillators move differently.

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For example, for the middle frequency here, which is square root of two, which would correspond to

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this, I can I convey a value here which you still have to calculate the square root.

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This would correspond to the second factor here, which is this one.

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And it indicates that only the first and the third oscillator, they move and they move along the same

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direction and the middle oscillator doesn't move at all.

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This is probably because that's a mode where these two oscillators here, they both move and they move

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in such a way that they leave this point in variance.

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And this is a preferred mode of isolation of the system.

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So you see the other two modes with the other frequencies are a bit more difficult because here are

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all three oscillators move and they move also along opposite directions partially.

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And you see that the middle oscillator moves a bit stronger, has a higher amplitude because a 0.7 is,

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of course, larger than 0.5.

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So in the following two lectures, we will use a four year transform and we will use a fit methods to

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figure out how we can find these IGen frequencies that we have determined that we can identify them

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with our numerical solution.

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And after these two lectures, we will see that this pretty chaotic behavior here is, of course, not

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really chaotic from a mathematical sense.

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But this difficult looking behavior here, which is not at all harmonic, can be represented by the

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superposition of three harmonic sine function, for example.

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So these are just two superposition of three sine functions, and you may guess it already, since these

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are the EIGEN frequencies.

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These will be the frequencies for the three periodic functions, the three sine functions.

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We will first analyze this by calculating the year transform of the spectrum here, and then later on,

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we will establish a fit that will fit the three functions to the numerically generated data.

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And then we really know what the numerical solution means.