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Eigensystem ignoring assumptions


I am trying to understand, why Eigensystem is ignoring the assumptions and
still dividing by something that is explicitly assumed to be zero. This is
a more extreme counterpart of the b>=0 case. Any comments or references to
the documentation are appreciated.

In[320]:= Clear["Global`*"]

In[321]:= $Assumptions = b == 0;

In[322]:= m1 = ({
    {a, b},
    {b, -a}

In[323]:= Eigensystem[m1]

Out[323]= {{-Sqrt[a^2 + b^2], Sqrt[
  a^2 + b^2]}, {{-((-a + Sqrt[a^2 + b^2])/b),
   1}, {-((-a - Sqrt[a^2 + b^2])/b), 1}}}

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