TITLE: Importance of eigenvalues
QUESTION [19 upvotes]: I know how to find eigenvalues and eigenvectors.  But I don't know what to do with that. 
What is their use?  Can anyone explain it to me?

REPLY [5 votes]: As far as I'm concerned, the biggest deal about eigen-anything is that there are many canonical cases where the eigen-vectors/functions/states form a complete basis for the space that you're working in. For instance, in the case of Quantum Mechanics, the "eigenvectors" of operators which corresponds to observable quantities, like energy, position, etc., form a complete basis for the space of all possible states of the system that you're analysing. That is, any state you want can be written as a linear combination of these "eigenvectors". Naturally, this makes solving problems so much easier, as now all we need to do is to find the coefficients in this linear combination (for which there is a neat formula). 
Many mathematics techniques such as Fourier Analysis work on the same principle. There is also a technique for solving certain types of linear ODEs which exploits the fact that the "eigenvectors" of certain things for a complete basis (Sturm-Liouville Theory), basically what you do is to find the "eigenvectors" of the differential equation (strictly, we want the eigenfunctions of the differential operator, e.g. $e^{kx}$ is an eigenfunction of $\frac{\mathrm{d}}{\mathrm{d}x}$ with eigenvalue $k$) and then we can write the solution in terms of these "eigenvectors" simply by calculating the coefficients in the linear combination (there is a formula for this too!).