We're making a video presentation on the topic of eigenvectors and eigenvalues. Unfortunately we have only reached the theoretical part of the discussion. Any comments on practical applications would be appreciated.
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The problem of ranking the outcomes of a search engine like Google is solved in terms of an invariant measure on the net, seen as a Markov chain. Finding the invariant measure requires the spectral analysis of the associated matrix. 


I would comment on Peitro's answer, but I don't have enough reputation; for a marvelouslytitled explanation of Google's Pagerank, see The $25,000,000,000 Eigenvector. 


Google's pagerank system is most likely the most canonical example, however others include, Dynamical System If you are able to express a model in terms of a matrix acting on vectors, one can look at the iterations and ask what occurs? This can be done to model the life cycle of some species in an environment (bacteria on a petri dish, wolf/sheep interaction, fibonacci sequence as the spread of a population of bunnies, etc...). These examples are fairly small, however you can certainly have massive systems to model, and if your matrix is diagonalizable, the iterations of this map correspond to iterations of a diagonal matrix (very easy to do!) instead of the standard $m^{2}$ operations to multiply out an $m\times m$ matrix. Think about a $1 000 000 \times 1 000 000$ matrix $M$, where you're looking at whether a certain species will die out (i.e., itererating $M^{n}$ and checking as $n\to\infty$. Quite the time saver!) Graph theory As an undergrad one of my summer research projects looked into special graphs called (3,6)fullerenes, where we were finding that, looking at the adjacency matrix of the graph, one could pick 3 well chosen eigenvalues and their corresponding eigenvetors, and generate nice 3d plots of the graphs, whereas other choices would produce degenerate images, involving some twisted 2d surface. Differential equations One can use eigenvalues and eigenvectors to express the solutions to certain differential equations, which is one of the main reasons theory was developed in the first place! I would highly recommend reading the wikipedia article, as it covers many more examples than any one reply here will likely contain, with examples along to way! (Schrödinger equation, Molecular Orbitals, Geology and Glaciology, Factor Analysis, Vibration Analysis, Eigenfaces, Tensor of Inertia, Stress Tensor, Eigenvalues of a Graph) 


All of Quantum Mechanics is based on the notion of eigenvectors and eigenvalues. Observables are represented by hermitian operators Q, their determinate states are eigenvectors of Q, a measure of the observable can only yield an eigenvalue of the corresponding operator Q. If you measure an observable in the state $\psi$ in a system and find as result the eigenvalue $a$, the state of the system just after the measurement will be the normed projection of $\psi$ onto the eigenvector associated to $a$. And so on and so forth. Of course Quantum Physics is not mathematically trivial: the arena is infinite dimensional Hilbert Space (or more complicated functional analytic structures like Gelfand triples), operators are not bounded, etc...However, in the extremely fast growing field of Quantum Computing the algebra is mostly limited to finitedimensional spaces and their operators. Finally, let me mention that Frank Wilczek, a winner of the 2004 Nobel Prize in Physics, has interestingly reminisced that as a student he found Quantum Mechanics easier than Classical Mechanics because of its nice axiomatization alluded to above.. 


For visual appeal, you should look into the area of pendulums. There is a good demonstration with swinging bottles, I recall, and this does depend on eigenvalues that are nearly equal. Do a Web search on "coupled pendulums". 


Principal Component Analysis is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. It is very difficult to visualize data in high dimensional space, but PCA can be used their to analyze data. From the data set covariance matrix is formed and then eigen values and eigen vectors of that covariance matrix are found. These eigne values and eigen vectors then can be compared to figure out the contribution of a particular feature in the data set. Thus PCA can be successfully applied to reduce dimension of the data. 


In telecommunications the socalled "beamforming" algorithm in case of multiple antennas requires calculation of eigenvectors. 


I think the book $Spectra$ $of$ $Graphs$$:$ $Theory$ $and$ $Applications$ by Dragos M. Cvetkovic, Michael Doob, Horst Sachs and M. Cvetkovic is very good source for practical applications of eigenvalues and eigenvectors. In communication theory, coding theory and cryptography, the minimum distance of codes is very important parameter in decoding and also is very important in coding based cryptography (for example McEliece cryptosystem). It is interesting that the second largest eigenvalue of related graph to a code, can determine a good lowerbond for minimum distance of code. 


Another interesting application is rigid body rotation theory. No matter how complicated an object looks, there's always (at least) a set of three mutually orthogonal directions around which it can rotate perfectly without precession. Maybe not something you can base a whole lecture on, but it's a nice remark. 

