yes a matrix can have 0 as an eigenvalue. if, however, 0 is the DOMINANT eigenvalue, this means that 0 is the ONLY eigenvalue. since A is assumed diagonalizable, this means there is an invertible matrix P for which:

A = P0P^{-1}= 0.

if A = 0, then EVERY vector is a dominant eigenvector, so we get one on the first iteration (pick any non-zero x_{0}. whoa! a dominant eigenvector!). this is a trivial case, and can be disregarded. so we can assume that A has a non-zero eigenvector.

there do exist non-zero matrices with only 0 eigenvalues, but these matrices are not diagonalizable (such matrices are called nilpotent). here is an example: A =

[0 1]

[0 0].

let me try to give you an idea of what something like theorem could be used for. imagine you have an incoming sound-signal, composing of various component wave-forms. you have a signal processor, which acts as a linear operator. an eigenvector in this case, is a wave-shape which is amplified with minimal distortion (theoretically 0). therefore if you arrange that the sound signal input is matched to the eigenvectors of the processor, you will get a clean sound, but with perhaps say the treble boosted. technology related to this is actually used in noise filters, to sort out signals that carry useful information from "scattered" (non-eigenvector input) wave-forms.

similar technology is used to make cars quieter on the road (inside the cabin), or test a steel beam for imperfections (the beam is struck by a hammer, and the eigenvalues are "heard". an experienced steel worker knows what good eigenvalues sound like-they listen for the dominant eigenvalue, which should have a clear, crisp bell-like tone).

determining the largest (dominant) eigenvalue is key to the speed and efficiency of search engines like google. when that low-rider car with the blow-out speakers drives by and shakes the street...yes, you can blame eignevectors. systems of differential equations often have linear models....and what matters in these models is the size of the eigenvalues. even if these systems are "chaotic" at dominant eigenvalues they often display coherent and predictable behavior.

often, some underlying symmetry in the modelling of a situation justifies the assumption the the matrix we assign to it is diagonalizable. diagonalization is GOOD, because diagonal matrices are much easier to compute with than arbitrary ones (fewer computations). if we already know a matrix is diagonalizable, but haven't actually diagonalized it (computing the inverse of an nxn matrix is not a user-friendly task, for n > 6, let's say), computing the eigenvectors numerically is sometimes useful (especially if you just want "the dominant eigenspace"). remember the matrices you will do problems with in your text are "toy matrices" like 2x2 or 3x3, for the most part. in the "real world" matrices can have 100's or thousands of rows/columns.