Trac 18040: implement minimal polynomial for matrices over the symbol…

…ic ring

src/sage/interfaces/maxima_lib.py

@@ -141,7 +141,7 @@
 # keepfloat -- don't automatically convert floats to rationals
 init_code = ['display2d : false', 'domain : complex', 'keepfloat : true',
             'load(to_poly_solve)', 'load(simplify_sum)',
-            'load(abs_integrate)']
+            'load(abs_integrate)', 'load(diag)']
 
 # Turn off the prompt labels, since computing them *very
 # dramatically* slows down the maxima interpret after a while.

src/sage/matrix/matrix2.pyx

@@ -1924,12 +1924,6 @@ cdef class Matrix(matrix1.Matrix):
             sage: factor(A.minpoly('y'))
             (y + 1) * (y + 2)^2
 
-        We can take the minimal polynomial of symbolic matrices::
-
-            sage: t = var('t')
-            sage: m = matrix(2,[1,2,4,t])
-            sage: m.minimal_polynomial()
-            x^2 + (-t - 1)*x + t - 8
         """
         f = self.fetch('minpoly')
         if not f is None:

src/sage/matrix/matrix_symbolic_dense.pyx

@@ -148,7 +148,7 @@ Conversion to Maxima::
 
 """
 
-
+from sage.rings.polynomial.all import PolynomialRing
 from sage.structure.element cimport ModuleElement, RingElement, Element
 from sage.structure.factorization import Factorization
 
@@ -451,6 +451,39 @@ cdef class Matrix_symbolic_dense(matrix_generic_dense.Matrix_generic_dense):
         self.cache(cache_key, cp)
         return cp
 
+    def minpoly(self, var='x'):
+        """
+        Return the minimal polynomial of ``self``.
+
+        EXAMPLES::
+
+            sage: M = Matrix.identity(SR, 2)
+            sage: M.minpoly()
+            x - 1
+
+            sage: t = var('t')
+            sage: m = matrix(2, [1, 2, 4, t])
+            sage: m.minimal_polynomial()
+            x^2 + (-t - 1)*x + t - 8
+
+        TESTS:
+
+        Check that the variable `x` can occur in the matrix::
+
+            sage: m = matrix([[x]])
+            sage: m.minimal_polynomial('y')
+            y - x
+
+        """
+        mp = self.fetch('minpoly')
+        if mp is None:
+            mp = self._maxima_lib_().jordan().minimalPoly().expand()
+            d = mp.hipow('x')
+            mp = [mp.coeff('x', i) for i in xrange(0, d + 1)]
+            mp = PolynomialRing(self.base_ring(), 'x')(mp)
+            self.cache('minpoly', mp)
+        return mp.change_variable_name(var)
+
     def fcp(self, var='x'):
         """
         Return the factorization of the characteristic polynomial of self.