TITLE: What is the mechanism of Eigenvector?
QUESTION [6 upvotes]: I have studied EigenValues and EigenVectors but still what I can't see is that how EigenVectors become transformed or rotated vectors.

REPLY [2 votes]: Linear maps $T:\>V\to W$ make sense between any two vector spaces $V$, $W$ over the same ground field $F$.
The notions of eigenvalues and eigenvectors only make sense for linear maps $T:\>V\to V$ mapping some vector space $V$ into itself. A vector $v\ne0$ that is "by coincidence" mapped onto a scalar multiple $\lambda v$ of itself is called an eigenvector of $T$. Such a vector is maybe shrinked or stretched by $T$, but it is not "rotated" or "sheared" in any way. The numbers $\lambda$ occurring in such circumstances are called eigenvalues of $T$. It is a miracle that (in the finite dimensional case) a given $T$ may only have a finite number of eigenvalues.