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This is a question about an off-hand remark from a lecturer a few weeks ago. He was talking about eigenvectors/values of a matrix, and rhetorically asked us if we'd seen the interpretation of eigenvalues as frequencies.

I haven't been able to find a clear explanation for this, and I'd be interested if anyone could enlighten me or (perhaps even more usefully) point me in the direction of a good text which covers this interpretation/application.

Thankyou very much!

tom
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  • It only makes sense if your vector space can reasonably be interpreted as modelling oscillations. Otherwise, I don't see why you would gain something by speaking about frequencies instead of eigenvalues. Besides, even when they can be interpreted as frequencies, we still speak about eigenfrequencies of the system to distinguish them from frequencies which are no eigenvalues of the operator under consideration. – Raskolnikov Apr 02 '12 at 10:11

2 Answers2

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Standard Fourier series introductory material. The linear transformation (instead of a matrix) is $$ T = \frac{-d^2}{dx^2}. $$ The vector space is smooth functions of $x$ with period $2\pi.$ And we get $$ T (\cos n x ) = n^2 \cos n x,$$ so $n^2$ is an eigenvalue of $T.$ However, $$ T (\sin n x ) = n^2 \sin n x$$ as well, so we get two different eigenvectors for that eigenvalues. How different are they? Sticking with real functions, we have an inner product $\langle, \rangle$ on pairs of periodic functions given by $$ \langle f,g \rangle = \int_0^{2 \pi} \; f(x) g(x) dx. $$ And the pair of functions we gave are orthogonal under the given inner product. Also other pairs, such as $\cos nx, \cos mx$ give $0$ when $m \neq n,$ same for sines, same for sin and cosine, one $m$ the other $n.$ Finally, integration by parts tells us that $T$ is self-adjoint with respect to the inner product, as $$ \int_0^{2 \pi} \; u'(x) v'(x) dx = \langle u,Tv \rangle = \langle Tu,v \rangle. $$

Will Jagy
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  • I can't get the exact mean of the first formula. Can you describe it a little bit more? Do you mean that applying the Fourier transform on a function is like finding the second derivative of that function? Thanks – user137927 Jun 11 '20 at 19:38
  • Dear Will, This is a great answer, but I couldn't find any reference that shows the standard Fourier series is equal to the second derivative. Would you please give me a reference for more information? – user137927 Sep 17 '20 at 11:48
  • @user really had not intended to discuss infinite series of any kind. I seem to have meant "standard fourier series intoductory material." In brief, what I wrote is adequate for finite fourier series, which are linear combinations of $1, \cos nx ; , ; \sin nx ; . ; $ The vector space would be linear combinations with $n \leq B$ where $B$ is a positive integer. We get lots of questions on linear algebra of ordinary one-variable polynomials with degree up to some $B,$ sometimes including Gram Schmidt. I gave an answer on such within the past month, let me try to find it. – Will Jagy Sep 17 '20 at 12:55
  • @user https://math.stackexchange.com/questions/3818087/gram-schmidt-method-to-get-a-basis-for-p-3/3818664#3818664 – Will Jagy Sep 17 '20 at 13:04
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I'd say that this is not a general way to think about eigenvalues, but rather an observation that when solving certain differential equations we can use the eigenvector and eigenvalue concepts, and they correspond to frequency in this case. This turns out to be really slick and useful, and so mathematicians, who will grab any tool and use it in any possible situation until it completely breaks, will use this as an analogy and start describing other quantities using the language of waves. This ends up being useful, but it also ends up with mathematicians using common language in uncommon and highly generalized ways.

As for learning about this, I'd start with a Google search on "separation of variables wave equations eigenfunction approach". The first result I get is a nice PDF to start you out; others may be able to suggest reading that take it further.

JonathanZ
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