Eigen Vectors and Basis!

Looks like, I'm on a roll with Dr. Maths. He's explained Eigen vectors this time, and trust me this is the best one, so far I have come across.

Source: http://mathforum.org/library/drmath/view/51971.html

I will give you a physical description, i.e. using 2 or 3 dimensions
only, though the ideas can be extended to n dimensions where n is as
big as you please.

You will be aware that if say a (2x2) matrix M operates on a two-
dimensional column vector v, then that vector is transformed in
magnitude or direction or both to the vector v'.

    So   M.v = v'

Now for the general (2x2) matrix M there are 2 eigenvalues k1 and k2
with associated eigenvectors u1 and u2 with the property that:

        M.u1 = k1.u1

        M.u2 = k2.u2

So any point on the vector u1 is transformed to k1.u1 when operated
upon by M, and similarly any point on u2 will move to k2.u2 after
transformation by M.  In some problems where M is to transform a
complicated figure or we wish to describe the transformation clearly,
it is convenient to use u1 and u2 as the base vectors - i.e. give
coordinates of all points in terms of u1 and u2 rather than the usual
(x,y) coordinates, and the transformation matrix then becomes

        |k1    0|
        |0    k2|

We can find powers of matrices very conveniently using eigenvalues and
eigenvectors.  It is easy to show that

  M = (u1 u2)|k1   0|(u1 u2)^(-1)
             |0   k2|

where (u1 u2) is the 2x2 matrix P formed by the columns of u1 and u2.

 Then M^n = P|k1   0|^n P^(-1)
             |0   k2|

      M^n = P|k1^n    0|P^(-1)
             |0    k2^n|

In probability theory, powers of matrices are frequently required,
sometimes infinite powers, so some device for handling such a problem
is clearly very important.

Again, mind-blowing view of something that we hear daily. In short, if you choose your basis to represent a vector as the eigen vectors of the transformation (or Matrix) then working with the new transformation (or matrix) and new vector is simple and straight forward!