Moment of Inertia is a number no more: The quest for the meaning of Eigenvalues begins!

It turns out the subject of this entry is not just to attract eyes, It is true!!! Well, not entirely (There's always a catch). A few weeks ago, I decided I will just 'sit in' an analytical mechanics class and see what all the physics and applied math majors were upto. So, it turned out I was the only Electrical engineering student in the class (No typos there) and I was proud! Ok enough about me now.
So, due to the unanswered questions about the eigenvalues (ref previous entry), I dug them up in various places after that first Linear Algebra lecture 3 semesters ago. The first one being Quantum Mechanics, then my own branch (Electrical Engineering), then in Mechanics!! and later even in fundamental physical quantities in Electrodynamics!! Anyway, I decided I'll write what I know about the eigenvalues I encountered in the most "classical" of the physics branches first. So, the events occurred in the following fashion that day: I was half asleep as usual while the professor was generalizing and over-generalizing and hyper-generalizing (by which time, I completely lost track of the thing that we were attempting to generalize ...psst...It was angular momentum....I think...anyway something related to it at least), then I began seeing some truly horrendous expressions on the board which would have made a much gentler impact on my brain if the prof used the "Einstein summation convention" (Just a name given to the process that makes huge sums look simple). So, while I was receiving the impact of these expressions head on, the words "...So, students, Moment of Inertia is not a number anymore! It turns out that it is a 3x3 matrix in this case, which is a special case of what we call a TENSOR....." fell in my ears and I got very excited about what he was saying (It happens when you don't know something and in the middle of something else that is familiar, that something, that you don't have any clue about, pops up! and the something that I did not know and regrettably, still don't was the meaning of the word TENSOR) So, naturally, by this time I was more attentive when what could almost be termed as magic unfolded in front of me! The professor was great and he was excited about it too. Then, we went on, calculated the "Principal Axes" of the system using the eigenvalues of the moment of inertia tensor. So, on that fateful day, I came across a whole new way of looking at eigenvalues. (May be I'll upload a document/pdf later showing the exact derivation done in class).