Trac #12595: Add doctest for eigenmatrix of complex floating-point ma…

…trix

{{{
sage: m = Matrix(CDF, 8, [[-1, -1, -1, -1, 1, -3, -1, -1], [1, 1, 1, 1,
-1, -1, 1, -3], [-1, 3, -1, -1, 1, 1, -1, -1], [-1, -1, -1, 3, 1, 1, -1,
-1], [1, 1, -3, 1, -1, -1, 1, 1], [1, 1, 1, 1, -1, -1, -3, 1], [3, -1,
-1, -1, 1, 1, -1, -1], [1, 1, 1, 1, 3, -1, 1, 1]])
sage: d, p = m.eigenmatrix_left()
sage: (p[1] * m)[0] / p[1][0]
1.2360679775 - 3.80422606518*I
sage: d[1][1]
1.2360679775 + 3.80422606518*I
}}}
Sage seems to return the complex conjugate of {{{d}}} or something of
the sort. Perhaps {{{d}}} is simply wrongly permuted (but real
eigenvalues seem to be correct).

{{{p.inverse() * d * p}}} should be at least approximately equal to
{{{m}}}.

This was reported on
[https://spreadsheets.google.com/pub?key=pCwvGVwSMxTzT6E2xNdo5fA the
public bug reports from the notebook interface] by
<[email protected]> on 1/24/2012.

URL: https://trac.sagemath.org/12595
Reported by: dkrenn
Ticket author(s): Shashwat Singh
Reviewer(s): Samuel Lelièvre

src/sage/matrix/matrix2.pyx

@@ -7000,6 +7000,21 @@ cdef class Matrix(Matrix1):
             Traceback (most recent call last):
             ...
             RuntimeError: failed to compute eigenvectors for eigenvalue ..., check eigenvectors_left() for partial results
+
+        The following example shows that :trac:`12595` has been resolved::
+
+            sage: m = Matrix(CDF, 8, [[-1, -1, -1, -1, 1, -3, -1, -1],
+            ....:                     [1, 1, 1, 1, -1, -1, 1, -3],
+            ....:                     [-1, 3, -1, -1, 1, 1, -1, -1],
+            ....:                     [-1, -1, -1, 3, 1, 1, -1, -1],
+            ....:                     [1, 1, -3, 1, -1, -1, 1, 1],
+            ....:                     [1, 1, 1, 1, -1, -1, -3, 1],
+            ....:                     [3, -1, -1, -1, 1, 1, -1, -1],
+            ....:                     [1, 1, 1, 1, 3, -1, 1, 1]])
+            sage: d, p = m.eigenmatrix_left()
+            sage: mm = p.inverse() * d * p  # This should be equal to m up to numerical noise
+            sage: norm(mm - m) < 1e-12
+            True
         """
         from sage.misc.flatten import flatten
         from sage.matrix.constructor import diagonal_matrix, matrix