Eigenvalues/vectors as measures of 'frequency'
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!
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$\begingroup$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$$ \int_0^{2 \pi} \; u'(x) v'(x) dx = \langle u,Tv \rangle = \langle Tu,v \rangle. $$
$\endgroup$ 4 $\begingroup$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.
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