Trac #7989: Minpoly doesn't work for all matrices

Right now, not all matrices can compute minpolys.  This patch exposes
these matrices.

URL: https://trac.sagemath.org/7989
Reported by: jason
Ticket author(s): Jason Grout, Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

src/sage/matrix/matrix2.pyx

@@ -2794,20 +2794,28 @@ cdef class Matrix(Matrix1):
 
             sage: factor(A.minpoly('y'))
             (y + 1) * (y + 2)^2
-
         """
         f = self.fetch('minpoly')
         if not f is None:
             return f.change_variable_name(var)
         f = self.charpoly(var=var, **kwds)
-        if f.is_squarefree():  # is_squarefree for polys much faster than factor.
-            # Then f must be the minpoly
-            self.cache('minpoly', f)
-            return f
+        try:
+            no_sq = f.is_squarefree()
+            # is_squarefree for polys much faster than factor.
+        except (TypeError, NotImplementedError):
+            pass
+        else:
+            if no_sq:
+                # Then f must be the minpoly
+                self.cache('minpoly', f)
+                return f
 
         # Now we have to work harder.  We find the power of each
         # irreducible factor that divides the minpoly.
-        mp = f.radical()
+        try:
+            mp = f.radical()
+        except (TypeError, NotImplementedError):
+            raise NotImplementedError("minimal polynomial not implemented")
         for h, e in f.factor():
             if e > 1:
                 # Find the power of B so that the dimension
@@ -2825,6 +2833,20 @@ cdef class Matrix(Matrix1):
         self.cache('minpoly', mp)
         return mp
 
+    def _test_minpoly(self, **options):
+        """
+        Check that :meth:`minpoly` works.
+
+        EXAMPLES::
+
+            sage: a = matrix([[1,2],[3,4]])
+            sage: a._test_minpoly()
+        """
+        if self.nrows() == self.ncols() and self.base_ring().is_exact():
+            tester = self._tester(**options)
+            # At least check that the minimal polynomial kills the matrix
+            tester.assertTrue(self.minpoly().subs(x=self).is_zero())
+
     def charpoly(self, var = 'x', algorithm = None):
         r"""
         Returns the characteristic polynomial of self, as a polynomial over

src/sage/matrix/matrix_dense.pyx

@@ -5,7 +5,7 @@ TESTS::
 
     sage: R.<a,b> = QQ[]
     sage: m = matrix(R,2,[0,a,b,b^2])
-    sage: TestSuite(m).run()
+    sage: TestSuite(m).run(skip="_test_minpoly")
 """
 
 cimport sage.matrix.matrix as matrix

src/sage/matrix/matrix_gap.pyx

@@ -416,6 +416,21 @@ cdef class Matrix_gap(Matrix_dense):
         """
         return int(self._libgap.RankMat())
 
+    def minpoly(self, var='x', **kwds):
+        """
+        Compute the minimal polynomial.
+
+        EXAMPLES::
+
+            sage: M = MatrixSpace(ZZ, 2, implementation='gap')
+            sage: M([0, 1, -1, -1]).minpoly()
+            x^2 + x + 1
+        """
+        po = self._libgap.MinimalPolynomial().sage()
+        return po.change_variable_name(var)
+
+    minimal_polynomial = minpoly
+
     def elementary_divisors(self):
         """
         Return the list of elementary divisors of this matrix.

src/sage/matrix/matrix_generic_dense.pyx

@@ -28,7 +28,7 @@ cdef class Matrix_generic_dense(matrix_dense.Matrix_dense):
         sage: A = random_matrix(Integers(25)['x'], 2)
         sage: type(A)
         <class 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
-        sage: TestSuite(A).run()
+        sage: TestSuite(A).run(skip='_test_minpoly')
 
     Test comparisons::