There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ones)


**My question:**  

What is the most helpful intuition to get a feeling for what this "eigen" really means (in its core)? What is the connecting element? And what examples would you give to clarify this concept?



Intuitions/connections/examples for "eigen-*"
When you see the word **eigen**, replace it with the term **spectrum of an operator** (see [spectral theory](http://en.wikipedia.org/wiki/Spectral_theory)) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^{-1}$) represent the same transform with respect to a different base.


When you diagonalize the matrix, you are actually trying to obtain an [orthogonal](http://en.wikipedia.org/wiki/Orthogonal_transformation) decomposition of the transform as a linearly independent eigensystem. 


1. If there are n independent
eigenvectors, you will obtain a full
diagonalization of your matrix.
2. If less than n, you have two choices. If all eigenvalues are in the ground 
field, you will get a [Jordan decomposition](http://mathworld.wolfram.com/JordanCanonicalForm.html). Otherwise, you have to settle with a [rational canonical form](http://mathworld.wolfram.com/RationalCanonicalForm.html).


In addition to Gilbert Strang's excellent book and lectures on Linear Algebra, I recommend browsing through Castillo's [Orthogonal sets and polar methods in linear algebra](http://books.google.com/books?id=KXKhIdVDMsAC&lpg=PP1&ots=DrsW5LrMeu&dq=castillo%2520linear%2520algebra&pg=PR9#v=onepage&q&f=false). Throughout the book, the matrix is seen as a transform rather than something which must be numerically manipulated. 


