A matrix is diagonalizable, so what?
Up to change in basis, there are only 2 things a matrix can do.
- It can act like a scaling operator where it takes certain key vectors (eigenvectors) and scales them, or
- it can act as a shift operator where it takes a first vector, sends it to a second vector, the second vector to a third vector, and so forth, then sends the last vector in a group to zero.
It may be that for some collection of vectors it does scaling whereas for others it does shifting, or it can also do linear combinations of these actions (block scaling and shifting simultaneously). For example, the matrix $$ P \begin{bmatrix} 4 & & & & \\ & 3 & 1 & & \\ & & 3 & 1 &\\ & & & 3 & \\ & & & & 2 \end{bmatrix} P^{-1} = P\left( \begin{bmatrix} 4 & & & & \\ & 3 & & & \\ & & 3 & &\\ & & & 3 & \\ & & & & 2 \end{bmatrix} + \begin{bmatrix} 0& & & & \\ & 0& 1 & & \\ & & 0& 1 &\\ & & & 0& \\ & & & &0 \end{bmatrix}\right)P^{-1} $$ acts as the combination of a scaling operator on all the columns of $P$
-
$p_1 \rightarrow 4 p_1$, $p_2 \rightarrow 3 p_2$, ..., $p_5 \rightarrow 2 p_5$,
plus a shifting operator on the 2nd, 3rd and 4th columns of $P$:
$p_4 \rightarrow p_3 \rightarrow p_2 \rightarrow 0$.
This idea is the main content behind the Jordan normal form.
Being diagonalizable means that it does not do any of the shifting, and only does scaling.
For a more thorough explanation, see this excellent blog post by Terry Tao: http://terrytao.wordpress.com/2007/10/12/the-jordan-normal-form-and-the-euclidean-algorithm/
If an $n\times n$ real matrix $M$ is diagonalizeable, then it corresponds to stretching $\mathbb R^n$ along $n$ linearly independent lines some factor $\lambda_n$. Otherwise this is not the case.
I'll try an answer in a different (equivalent) direction: what happens when the matrix is not diagonalizable?
First of all, this must mean that some of the matrix's eigenvalues occur more than once. Otherwise the matrix really can't do anything else than simply stretching its eigenvectors by $\lambda_n$. So what if two eigenvalues are equal? Let's write down the most non-trivial non-diagonalizable example there is $$\pmatrix{0 & 1 \cr 0 & 0}.$$ This (as you have correctly observed) crushes the space. But the important point is that it doesn't crush it to zero! Instead it only crushes it to some subspace. In general, if you have a nilpotent matrix (all eigenvalues vanish) there are many subspaces (of varying dimensions) to pick from and so many different ways to crush the space. The zero operator sends everything to zero immediately while it will take a nontrivial nilpotent matrix some (finite) time in a sense that for $T$ nilpotent, $T^n = 0$ for some $n$. In general, every nilpotent matrix is similar to a matrix that looks something like this $$\pmatrix{0 & 0 & 1 & 0 \cr 0 & 0 &1 &1 \cr 0 & 0 & 0 & 1 \cr 0 & 0 & 0 &0}$$ with some ones above the diagonal. The precise position of those ones (if there are any) determines which subspace is being crush to which and so on. You are very much encouraged to play with such matrices in $\mathbb R^3$ (where there are not too many possibilties how a nilpotent matrix can look like, but still enough to show what happens).
Having said all this, it would be a sin now not to mention Jordan decomposition. When studying a matrix $A$ you first find its eigenvalues and corresponding eigenspaces. So pick such an eigenspace corresponding to the eigenvalue $\lambda$. Then $A - \lambda$, restricted to this eigenspace, is a nilpotent operator! If this nilpotent operator is zero, then the original matrix $A$ just stretches this eigenspace by $\lambda$. But in general it can perform a lot of nontrivial shuffling (corresponding to the nilpotent part).
Diagonalization is shearing any source of origin the x directional and y directional matric elements of rows and column from a source of for example radiation of light rays into squeezing along hypotenuse out of a right angled triangle sides as such resultant a+ib module of shearing.This happens as a result of stretching.It really bends a light ray by its refractive index along input and output planes.On the line of thought this may play the role of light ray invisibility by the Einstein gravity of bending typically applicable in Quantum mechanics.A sort of squeezing or shearing along hypotenuse side.When all the elements are diagonally shifted the potential becomes zero and not so when more than zero above the diagonalization of directional matrics elements.This may also be called a twisting along axial planes as a function of twisting angle of ray transfer matrics pave the way for magnification as divergence ,convergence as well as for invisible clocking dynamics using laser beams.