Applying a polynomial to a matrix

edit

FWIW :

sage: f(x) = x^2 + 5*x+3

Note : this is a symbolic function",more properly called a *callable symbolic expression whose components must be symbolic expressions.

sage: B=matrix([[1/2,2/3],[1/2,1/3]])  ; B
[1/2 2/3]
[1/2 1/3]

Note : B is a matrix, which is not a symbolic expression. Demonstration

sage: f(B)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File /usr/local/sage-9/src/sage/symbolic/expression.pyx:3804, in sage.symbolic.expression.Expression.coerce_in()
   3803 try:
-> 3804     return <Expression?>z
   3805 except TypeError:

TypeError: Cannot convert sage.matrix.matrix_rational_dense.Matrix_rational_dense to sage.symbolic.expression.Expression

Q. E. D.

Explanation later in the same sequence :

During handling of the above exception, another exception occurred:

TypeError                                 Traceback (most recent call last)
Cell In [76], line 1
----> 1 f(B)

File /usr/local/sage-9/src/sage/symbolic/expression.pyx:6182, in sage.symbolic.expression.Expression.__call__()
   6180         z^2 + x^y
   6181     """
-> 6182     return self._parent._call_element_(self, *args, **kwds)
   6183 
   6184 def variables(self):

File /usr/local/sage-9/src/sage/symbolic/callable.py:476, in CallableSymbolicExpressionRing_class._call_element_(self, _the_element, *args, **kwds)
    474 d = dict(zip([repr(_) for _ in self.arguments()], args))
    475 d.update(kwds)
--> 476 return SR(_the_element.substitute(**d))

File /usr/local/sage-9/src/sage/symbolic/expression.pyx:5891, in sage.symbolic.expression.Expression.substitute()
   5889 for k, v in sdict.iteritems():
   5890     smap.insert(make_pair((<Expression>self.coerce_in(k))._gobj,
-> 5891                           (<Expression>self.coerce_in(v))._gobj))
   5892 res = self._gobj.subs_map(smap, 0)
   5893 return new_Expression_from_GEx(self._parent, res)

File /usr/local/sage-9/src/sage/symbolic/expression.pyx:3806, in sage.symbolic.expression.Expression.coerce_in()
   3804         return <Expression?>z
   3805     except TypeError:
-> 3806         return self._parent.coerce(z)
   3807 
   3808 cpdef _add_(left, right):

File /usr/local/sage-9/src/sage/structure/parent.pyx:1213, in sage.structure.parent.Parent.coerce()
   1211         except Exception:
   1212             _record_exception()
-> 1213     raise TypeError(_LazyString("no canonical coercion from %s to %s", (parent(x), self), {}))
   1214 else:
   1215     return (<map.Map>mor)._call_(x)

TypeError: no canonical coercion from Full MatrixSpace of 2 by 2 dense matrices over Rational Field to Callable function ring with argument x

But (proper, Python) functions of B are possible :

sage: def g(x): return x^2+5*x+3
sage: g(B)
[73/12  35/9]
[35/12  46/9]

Q. E. D. again...

Unless you have meant :

sage: B.apply_map(lambda u:f(u))
[23/4 61/9]
[23/4 43/9]

which is something else entirely...

HTH,