What does eigenvalues and eigenvectors mean?

[chubbypaddy]

I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how.

What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways?
I know if you have linear transformation then the eigenvectors are those vectors that are just scaled and not changed in direction.

I think PageRank does something with eigenvectors but I don't know how, what's the idea?

Some good(simple though) text problems to actually model something real would be helpful.

 

[homeomorphic]

I suppose maybe what you are missing is the idea of thinking of a function as a vector. You can multiply functions by scalars, which stretches them (or shrinks), and you can add functions together. These are just the algebraic properties that vectors have. The only real difference is that with functions, there are lots of different linearly independent functions, so they are really vectors in an infinite dimensional space, usually. Unless, you have a function that is defined on a finite set of points. Then, you get a finite-dimensional vector space of real-valued functions.

Typically, in physics, you'll have a physical system and you can express it in terms of "normal modes", which are eigenvectors. Each normal mode oscillates at some frequency. The frequency is basically an eigenvalue corresponding to that eigenvector (normal mode). As an example, if you have a vibrating string with fixed end points, then the normal modes will be sine waves, like sin {n π) or something like that (under suitable assumptions), times some other factor that varies with time. Each sine wave is basically a vector. The significance of these normal modes is that they don't interact. So, rather than having a complicated system of interacting parts, you can just reduce it down to studying the motion of each normal mode. In other words, it's basically just diagonalizing a matrix. So, you can solve the system of differential equations by just solving one first order equation for each normal mode. What I have described is essentially the theory of coupled harmonic oscillators. You can approximate any system near an equilibrium by couple harmonic oscillators, although, general physical systems can get more complicated. These kinds of ideas show up in quantum mechanics and quantum field theory, as well as in the classical theory of waves and oscillations.

In quantum mechanics, the eigenvalues are possible values of an observable and the eigenvectors are, again, physical states, which could be described by a compex-valued wave function, for example.

 

[AlephZero]

I guess the main part of your problem is not knowing how to model "real world" systems to get some equations in matrix form. Here's a simple example of a vibrating spring-and-mass system: http://lpsa.swarthmore.edu/MtrxVibe/EigApp/EigVib.html

Modelling the vibration of something like a bridge uses the same principle, but a real-world model may have thousands (or even hundreds of thousands) or variables, and creating the matrices is done numerically (with a computer) not algebraically.