Confused with Eigenvalues and Eigenvectors and Vector transformations Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here: 
I am actually performing automatic operations on finding them, but I don't really understand what they are and what they are used for.
Those operations really look like transformations. What is the difference of transforming a matrix and finding its Eigenvectors and Eigenvalues anyway?
Thank you
 A: There seems to be some rather deep confussion here and I'm not sure from where to start...so just let's:
(1) Eigenvectors are not "produced" by a matrix. They are vectors that fulfill a very precise relation wrt an operator/square matrix .
(2) Eigenvalues don't "scale up" transformations/matrices, whether "transformed" or not (what this means in this context). Eigenvalues are scalars that saisfy a certain polynomial equation very closely related to a trasnformation/matrix
(3) Eigenvectors are not trasnformations. Read (1) above.
Eigenvalues/eigenvectors are names proceeding from the german "eigen", meaning (its) "own" or "self", "inherent or proper", etc.
A: I like to imagine a plane of arrows (vectors), which all are mulptiplied by a matrix. All the arrows will tranform and become new arrows (vectors) on the plane. Those arrows which keep their initial direction are eigenarrows and  eigenvalues determine how the arrows scale (shrink or strech). A negative eigenvalue would mean the arrow changes its direction to the opposite it had. All the other arrows on the plane tranform differently:  they are skewed and scaled. 
