Somewhat related to some of the questions that have been raised here:- 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.

 Wow... your comment is infact more useful pearl of wisdom for getting the intuition of Eigen-values/vectors. Can you suggest some book/reference that best conveys this intuition ? Somehow, Gilbert Strang's Linear Algebra does put me to sleep :/
 Linear algebra done right by Sheldon axlerLinear algebra that focuses on matrices of numbers is missing the point and a proper numerical linear algebra class will teach them better anyway
 I would recommend Down with Determinants! instead. Same author, and was the inspiration for that textbook.https://www.maa.org/sites/default/files/pdf/awards/Axler-For...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. :-)
 I recommend the Linear Algebra series of YouTube videos by 3blue1brown.
 >for the set of matrices that possess them ("transformation matrices that only perform stretch/contract"), eigenvectorsMinor nitpick: this should end with “real eigenvectors.” Rotation matrices certainly have eigenvectors, they’re just complex.

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