Large difference between algebraic and numerical eigenvector system

I’m working on a general procedure where I need to obtain the eigen-decomposition of a matrix. I noticed the following funny difference, though. When I use

A = {{0, 0, -(I/10000), 1/5760000000, 0, 0}, 
 {0, 0, -(I/10000), 0, 1/5760000000, 0}, 
 {-((7I)/250000), -((7 I)/250000), 0, 0, 0, 1/  16000000000}, 
 {150632/125, 11232/125, 0, 0,   0, -((7 I)/250000)}, 
 {11232/125, 150632/125, 0, 0, 0, -((7 I)/250000)}, 
 {0, 0, 1000, -(I/10000), -(I/10000), 0}}
{eval,evec}=Eigensystem[A]

I obtain a nice right eigenvector system with zeroes and real OR imaginary values on each column (I’m outputting the transpose of evec):
$$left(
begin{array}{cccccc}
frac{733 i}{802944000} & frac{1}{96000 sqrt{697}} & frac{733 i}{802944000} & -frac{1}{96000 sqrt{697}} & frac{i}{12304000} & frac{i}{12304000}
-frac{i}{22304000} & -frac{1}{96000 sqrt{697}} & -frac{i}{22304000} & frac{1}{96000 sqrt{697}} & frac{i}{12304000} & frac{i}{12304000}
-frac{1}{192000 sqrt{697}} & 0 & frac{1}{192000 sqrt{697}} & 0 & -frac{sqrt{33}}{24608000} & frac{sqrt{33}}{24608000}
-frac{769 i}{12 sqrt{697}} & -1 & frac{769 i}{12 sqrt{697}} & -1 & -frac{1}{769} left(36 i sqrt{33}right) & frac{36 i sqrt{33}}{769}
0 & 1 & 0 & 1 & -frac{1}{769} left(36 i sqrt{33}right) & frac{36 i sqrt{33}}{769}
1 & 0 & 1 & 0 & 1 & 1
end{array}
right)$$

However, if I run

{eval,evec}=Eigensystem[N[A]]

I obtain the following output

$$
left(
begin{array}{cccccc}
-text{1.2043417539000936$grave{ }$*${}^{wedge}$-7}-text{7.688193188799198$grave{ }$*${}^{wedge}$-9} i & text{1.8531222222424326$grave{ }$*${}^{wedge}$-7}-text{8.212513455384941$grave{ }$*${}^{wedge}$-9} i & text{3.3413281903381244$grave{ }$*${}^{wedge}$-7}-text{4.423526477148793$grave{ }$*${}^{wedge}$-24} i & -text{3.3413281903381207$grave{ }$*${}^{wedge}$-7}-text{2.930323256407006$grave{ }$*${}^{wedge}$-24} i & -text{7.743377801175729$grave{ }$*${}^{wedge}$-24}+text{7.596590615921973$grave{ }$*${}^{wedge}$-8} i & text{1.9865788552372406$grave{ }$*${}^{wedge}$-23}+text{7.596590615921986$grave{ }$*${}^{wedge}$-8} i
text{3.5233965561984676$grave{ }$*${}^{wedge}$-7}+text{3.7759202564356767$grave{ }$*${}^{wedge}$-10} i & -text{3.37554746682975$grave{ }$*${}^{wedge}$-7}+text{4.033430892140163$grave{ }$*${}^{wedge}$-10} i & text{3.4515872267481366$grave{ }$*${}^{wedge}$-8}-text{2.3326479805751402$grave{ }$*${}^{wedge}$-24} i & -text{3.451587226748133$grave{ }$*${}^{wedge}$-8}+text{2.374275106979798$grave{ }$*${}^{wedge}$-25} i & text{2.6645278093098886$grave{ }$*${}^{wedge}$-24}+text{7.596590615921924$grave{ }$*${}^{wedge}$-8} i & -text{3.0443234191254495$grave{ }$*${}^{wedge}$-24}+text{7.596590615921939$grave{ }$*${}^{wedge}$-8} i
-text{1.6614525879086802$grave{ }$*${}^{wedge}$-9}-text{5.270427857833774$grave{ }$*${}^{wedge}$-8} i & text{1.7747605189685852$grave{ }$*${}^{wedge}$-9}-text{3.4599580883513965$grave{ }$*${}^{wedge}$-8} i & -text{1.3028961283509568$grave{ }$*${}^{wedge}$-23}-text{8.378138931700507$grave{ }$*${}^{wedge}$-8} i & text{7.456547295202199$grave{ }$*${}^{wedge}$-24}-text{8.378138931700512$grave{ }$*${}^{wedge}$-8} i & -text{2.1819545346633222$grave{ }$*${}^{wedge}$-7}-text{2.4980389433541498$grave{ }$*${}^{wedge}$-23} i & text{2.1819545346633222$grave{ }$*${}^{wedge}$-7}-text{1.4102245048664485$grave{ }$*${}^{wedge}$-23} i
-0.274879-0.0204425 i & -0.449739+0.0218367 i & 0.895109, +0. i & 0.895109, +0. i & -text{1.0760380728691062$grave{ }$*${}^{wedge}$-18}-0.251361 i & text{1.1271449376910601$grave{ }$*${}^{wedge}$-17}+0.251361 i
0.923353, +0. i & 0.875453, +0. i & 0.135738, -text{4.836881707484031$grave{ }$*${}^{wedge}$-17} i & 0.135738, -text{2.5946706042721974$grave{ }$*${}^{wedge}$-17} i & -text{8.770004331448554$grave{ }$*${}^{wedge}$-17}-0.251361 i & text{1.3097277227809595$grave{ }$*${}^{wedge}$-16}+0.251361 i
-0.00842181-0.267155 i & -0.00899616+0.175383 i & -text{4.0179138893596894$grave{ }$*${}^{wedge}$-17}-0.424683 i & -text{6.433703220485387$grave{ }$*${}^{wedge}$-18}+0.424683 i & 0.934685, +0. i & 0.934685, +0. i
end{array}
right)
$$

The eigenvalues in each case are remarkably similar in magnitude, and in fact they really only differ in the sense that the second case has minimal non-zero imaginary parts… In the first case:
$$ left{-frac{sqrt{697}}{60000},-frac{sqrt{697}}{60000},frac{sqrt{697}}{60000},frac{sqrt{697}}{60000},-frac{sqrt{33}}{20000},frac{sqrt{33}}{20000}right} $$
in the second case
$$ {0.000440013, -text{1.7327019397255794$grave{ }$*${}^{wedge}$-21} i,-0.000440013+text{1.0587911840678754$grave{ }$*${}^{wedge}$-21} i,0.000440013, -text{9.237297459711376$grave{ }$*${}^{wedge}$-23} i,-0.000440013-text{2.4134262448873742$grave{ }$*${}^{wedge}$-21} i,-0.000287228-text{1.4321361106426905$grave{ }$*${}^{wedge}$-20} i,0.000287228, -text{3.049711151506121$grave{ }$*${}^{wedge}$-20} i} $$

Now, I understand that Mathematica has normalized the eigenvectors to give a ‘1’ at the bottom-most value in the algebraic case, and probably just normalized the norm in the numerical case. But I don’t understand what has happened to the 0 entries? I can’t spot anything in the numerical eigenvectors that is close to 0. In other words, it seems to me that theeigenvectors differ a lot from each other.