Getting a feel for the transformation A on vector x which lies outside of any eigenspace

Question

In one of his videos, after 13:25 Sal starts to talk about the interpretation of the eigenvectors and how they relate to a vector $x$ being transformed by the matrix $A$.

He then goes through showing what happens when $x$ would be in one of the eigenspaces.

My questions are:

1. what is the interpretation of the transformation being applied to a vector outside of any eigenspace, like any regular 3-dimensional $x$?
2. How could I "visualize" what happens to such a vector?
3. Is it related somehow to the characteristic polynomial? What is the meaning of the characteristic polynomial, beside giving us the eigenvalues at the intersections with the $x$-axis?

Answer 1

So long as your matrix is real and dimension is 2 or 3 then some software aid is available and in 2d you can do much of it by hand. You start with a unit circle in $xy$ plane and ask what does your matrix do to each point $(x,y)$ on circle. Your circle typically maps into an ellipse. Apps such as this do the work for you.

Answer 2

I didn't have a correct definition of eigenspaces. I'm not an expert on the last part, but I'll attempt to explain anyway.

1. If A has a limiting matrix $A^\infty$ (powers can't flip back and forth, for example), then for every (column) vector v, $A^\infty v$ is an eigenvector of A.

2. So the vectors that are not in any eigenspace tend towards the eigenspaces asymptotically.

3. "Meaning is not in things, but in between them." - Norman O. Brown. Matrices and polynomials both form rings. Rings are often studied geometrically by analogy with the polynomial ring, and how the 0-sets of polynomials behave best as curves, like x^2+y^2-1=0. Sending every matrix to its characteristic polynomial is a morphism between curves, called a regular map (as is sending it to that polynomial's discriminant, see this question), and the fact that it vanishes at every point of the n-by-n matrix algebra treated as the affine plane of dimension $n^2$ is proof of the Cayley-Hamilton theorem from a different perspective.