> Anyone who has taken linear algebra should know that [...]
My university level linear algebra class didn't touch practical applications at all, which was frustrating to me because I knew that it could be very useful to some background doing hobbyist game dev. I still wish I had a better understanding of the use cases for things like eigenvectors/values.
Here are some applications of eigenvectors and eigenvalues:
1) If you have a set of states and a stochastic transition function, which gives for each starting state, the probability distribution over what the state will be at the next time step, you can describe this as a matrix. The long-term behavior of applying this can be described using the eigenvectors and eigenvalues of this matrix. Any stable distribution will be an eigenvector with eigenvalue 1. If there is periodicity to the behavior, where for some initial distributions, the distribution will change over time in a periodic way, where it ends up endlessly cycling through a finite set of distributions, then the matrix will have eigenvalues that are roots of unity (i.e. a complex number s such that s^n = 1 for some positive integer n). Eigenvalues with absolute value less than 1 correspond to transient contributions to the distribution which will decay (the closer to 0, the quicker the decay.). When there are finitely many states, there will always be at least one eigenvector with eigenvalue 1.
2) Related to (1), there is the PageRank algorithm, where one takes a graph where each node has links to other nodes, and one models a random walk on these nodes, and one uses the eigenvector one (approximately) finds in order to find the relative importance of the different nodes.
3) Rotations generally have eigenvalues that are complex numbers with length 1. As mentioned in (1), eigenvalues that are complex numbers with length 1 are associated with periodic/oscillating behavior. Well, I guess it sorta depends how you are using the matrix. If you have a matrix M with all of its eigenvalues purely imaginary, then exp(t M) (with t representing time) will describe oscillations with rates given by those eigenvalues. exp(t M) itself will have eigenvalues that are complex numbers of length 1. This is very relevant in solutions to higher order differential equations or differential equations where the quantity changing over time is a vector quantity.
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But, for purposes of gamedev, I think probably the eigenvalues/eigenvectors are probably the less relevant things. Probably instead, at least for rendering and such, you want stuff like, "you can use homogeneous coordinates in order to incorporate translations and rotations into a single 4x4 matrix (and also for other things relating the 3d scene to the 2d screen)", and stuff about like... well, quaternions can be helpful.
Of course, it all depends what you are trying to do...