Trac #30518: eigenvectors over QQbar are incorrectly conjugated

Eigenvectors over QQbar could get incorrectly conjugated. In the
following example from [https://groups.google.com/g/sage-
devel/c/9Ss98-XKR9c devel], the eigenvectors of a matrix over
`QuadraticField(-1)` are computed in two ways, one of which incorrectly
returns the conjugate of some eigenvectors.

By converting the matrix to `QQbar` first:
{{{
sage: K.<i> = QuadraticField(-1)
sage: m = matrix(K, 4, [2,4*i,-i,0, -4*i,2,-1,0, 2*i,-2,0,0, 4*i+4,
4*i-4,1-i,-2])
sage: sorted([(e, matrix.column(vs)) for e, vs, _ in
m.change_ring(QQbar).eigenvectors_right()])
[(
    [  1   0]
    [1*I   0]
    [  0   0]
-2, [  0   1]
),
 (
                      [           1.000000000000000? + 0.?e-16*I]
                      [           0.?e-16 - 1.000000000000000?*I]
                      [           0.?e-16 - 6.605551275463989?*I]
-0.6055512754639893?, [1.000000000000000? + 1.000000000000000?*I]
),
 (
                    [           1.000000000000000? + 0.?e-17*I]
                    [           0.?e-17 - 1.000000000000000?*I]
                    [          0.?e-17 + 0.6055512754639893?*I]
6.605551275463989?, [1.000000000000000? + 1.000000000000000?*I]
)]
}}}

Supposedly, this result is correct. In contrast, computing the
eigenvectors directly, without first converting to `QQbar`, leads to
some eigenvectors getting conjugated:

{{{
sage: sorted([(QQbar(e), matrix.column(QQbar, vs)) for e, vs, _ in
m.eigenvectors_right()])
[(
    [1 0]
    [I 0]
    [0 0]
-2, [0 1]
),
 (
                      [                                        1]
                      [           0.?e-54 + 1.000000000000000?*I]
                      [           0.?e-53 + 6.605551275463989?*I]
-0.6055512754639893?, [1.000000000000000? - 1.000000000000000?*I]
),
 (
                    [                                        1]
                    [           0.?e-53 + 1.000000000000000?*I]
                    [          0.?e-52 - 0.6055512754639893?*I]
6.605551275463989?, [1.000000000000000? - 1.000000000000000?*I]
)]
}}}

Note that the `QuadraticField` constructor automatically sets an
embedding.
{{{
sage: K.coerce_embedding()
Generic morphism:
  From: Number Field in i with defining polynomial x^2 + 1 with i = 1*I
  To:   Complex Lazy Field
  Defn: i -> 1*I
}}}

URL: https://trac.sagemath.org/30518
Reported by: gh-mwageringel
Ticket author(s): Vincent Delecroix
Reviewer(s): Marc Mezzarobba

Author: Release Manager

src/sage/matrix/matrix2.pyx

@@ -6590,6 +6590,12 @@ cdef class Matrix(Matrix1):
             doctest:...: DeprecationWarning: "extend" should be used as keyword argument
             See https://trac.sagemath.org/29243 for details.
             []
+
+        Check :trac:`30518`::
+
+            sage: K.<i> = QuadraticField(-1)
+            sage: m = matrix(K, 4, [2,4*i,-i,0, -4*i,2,-1,0, 2*i,-2,0,0, 4*i+4, 4*i-4,1-i,-2])
+            sage: assert all(m*v == e*v for e, vs, _ in m.eigenvectors_right() for v in vs)
         """
         if other is not None:
             if isinstance(other, bool):