- for the set of matrices that possess them ("transformation matrices that only perform stretch/contract"), eigenvectors (with their associated eigenvalues) play a role quite analogous to the role primes play in the integer set. They provide a unique, identifying "spectrum" for said matrix. This is made explicit by eigendecomposition (spectral decomposition).
- with extension via singular value decomposition (SVD method) to any square matrix (e.g. "transformation matrices that might also shear, rotate"), certain operations such as exponentiation of the square matrix can performed very quickly once eigenvectors/eigenvalues have been obtained via the SVD method.
Linear algebra that focuses on matrices of numbers is missing the point and a proper numerical linear algebra class will teach them better anyway
As a graduate student at the time, it opened my eyes about a whole bunch of theorems that I thought I already knew. And it is a lot more approachable than a textbook.
It won an award for the best piece of expository writing on mathematics in 1996. It was a deserved win. :-)
Minor nitpick: this should end with “real eigenvectors.”
Rotation matrices certainly have eigenvectors, they’re just complex.