linear algebra - What if do NOT want Mathematica to normalize eigenvectors with Eigenvectors[N[matrix]]?

ORIGINAL QUESTION

Question speaks for itself. Mathematica normalizes the eigenvectors automatically when you type Eigenvectors[N[matrix]]. What if I do NOT want the vectors to be normalized? I looked around and found that N[Eigenvectors[matrix]] would in principle do the trick. I mean, that is the form I would like to have them expressed. However, I can't do it because I get this message (which by the way I don't fully grasp):

"Incorrect number 20 of eigenvectors for eigenvalue 1/2 (-1+Sqrt[5]) with multiplicity 2."

The size of the matrix is $2^8$ by $2^8$.

EDIT

I see that there's some confusion on what I am asking. Obviously, as any (nonzero scalar) multiple of an eigenvector is equivalent to the same eigenvector, my question is not making that much sense the way it is asked. Apologies for that, I asked in a rush. I will try to put in context what I am doing so that maybe it will be more clear what my point is.

I have a $2^8$ by $2^8$ matrix, call it $H$. I want to construct the matrix $Q$ which is the linear combination of the density matrices (projectors) coming from the eigenvectors of $H$ weighted by some coefficients $c_j$. In formula, this is:

$Q=\sum_{j=0}^{2^8}c_j v_j^T v_j$

where $v_j$ are the eigenvectors of $H$ and $v^T$ denotes the transpose of $v$. I do not want to specify the $c_j$'s, they remain (possibly complex) variables. I calculate the $v$'s using Eigenvectors[N[H]] because N[Eigenvectors[H]] is both too slow and ineffective (see the error message above).

My point is that when I am doing this, I have entries which are so close to zero that I am not able to recognize which zeroes are actually zeroes and which ones are just due to the precision of my calculation (let's call these ones "numerical zeroes").

How do I know this? I tried to make things simpler and did the same calculation for a smaller matrix, $2^5$ by $2^5$. This time N[Eigenvectors[H]] works well. However, when using Eigenvectors[N[H]] and N[Eigenvectors[H]] I get two very different matrices $Q$. That is of course expected, because of the normalization in Eigenvectors[N[H]]. But what got me was the fact that the number of zero entries in $Q$ is very different in the two cases, namely, there are much more zeroes when I use N[Eigenvectors[H]]. This is of course puzzling, seeing that Mathematica supposedly maintains the same order of the eigenvectors in the two cases. I suspect that that is due to what I called the "numerical zeroes". Chopping is not helping much, because say that one of your entries is of order $10^{-5}$ because of the rescaling of the normalization, then already you will have something of order $10^{-10}$ in $v^T v$. So in principle you don't know if you're chopping something meaningful or not.
So my idea was the following: if I can tell someway Mathematica to NOT normalize the eigenvectors of my matrix, the entries of the eigenvectors won't scale down to small numbers and I could easily distinguish between exact zeroes and "numerical" zeroes and I could Chop nicely.

Also what tells me that something is wrong is that when I take the commutator $[H,Q]$ I get zero in one case and non-zero in the other one. Of course, the right result should be zero by construction.

Another motivation is of course the following: the size of the output I get for $Q$ when using N[Eigenvectors[H]] is WAY less than when I'm using Eigenvectors[N[H]], because the amount of zeroes is much more. We're talking 800 kB vs 100 MB in the $2^5$ case. So when I try to construct $Q$ in the $2^8$ case it takes forever. And seeing that in the future I will need to increase the size exponentially up to $2^{12}$ or more, I have to fix this.

Hope this made it clearer.