Question regarding Eigenvalues and Eigenvectors and its importance When  I  apply  a  linear  transformation to  a  special  vector, this  vector doesn't  rotate  ,it  stays  in  the   same direction, but  it  can  get  scaled, the  special vector  is  an Eigenvector and  scaling (ie)  the  ratio  of  the size  of  it  after  applying  the  Linear  Transformation  A and before linear  transformation  is an Eigenvalue ,
But  this  stuff  seems  trivial , why is  knowing  about  a  vector  that doesn't  change  its  direction  after Linear  Transformation  so  important that  it  basically  appears everywhere, eg:  Quantum  Physics  uses  Eigenvectors  and  Eigenvalues  a  Lot
Is  there  a  deeper  meaning  to  this that  I  don't  see ?
 A: For one thing they're just very convenient mathematically. As the other comments said you can transform a matrix into its "jordan normal form" which for "nice" matrices is just a diagonal matrix. These diagonal matrices are very nice to handle in a lot of ways and make some difficult things rather easy: consider for example the matrix exponential that's important in e.g. differential equations. For diagonal matrices this is trivial to compute.
Furthermore you can use eigenvalues (or the "generalization" of them called singular values) to find out how a transformation stretches space which gives them geometric meaning.
As for why they often times crop up in applications aside from this: I think there are two reasons.
One is that they just are directly what we're interested in. Just consider some amplifier or oscilator circuit for example - it's clear that we are interested in it's eigenvalues and how we can tune them.
Another one is that a lot of maths is oriented towards turning hard problems into linear algebra problems, and once you have some endomorphism you might as well ask yourself "what are the eigenvalues of this? Can I interpret them in some way that'll tell me something new about my problem?" and because of "the unreasonable effectiveness of mathematics" this apparently works out quite nicely. In a way eivenvectors and eigenvalues encode the "essence" of a linear map, so it makes sense that they're useful and interesting; but the fact that they actually come up that often doesn't have a particularly deep reason I believe.
