# Symbolic matrices

asked 2011-04-12 01:01:26 -0500

Hi,

is it possible to define totally symbolic matrices. Something like

N,M = var('N,M')
a = matrix(SR, N, M)
b = matrix(SR, M, N)
c = a.dot(b)

So that c[i,j] = sum(a[i,k]*b[k,i],k,1,M) or do you really always need an non-symblic matrix size?

If you really always need a non-symblic matrix type, is it possible to define the symbolic variables automatically? Something like

a = matrix(SR, 2, 3)
print a[i,j]
output:
a_(i,j)

Thanks!

Manuel

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answered 2011-04-12 13:28:29 -0500

There is the Tensor module in sympy. If that is the type of thing that you're looking for (ie explicit indices).

The was some code put on the sage devel group last year: abstract matrices, which allows basic manipulation of abstract matrices and vectors. It's not complete, but it does the basics and should be easy to extend if you need more. As far as I can tell, it hasn't been worked on since the original postings. I hope it's ok if I post the code here:

from sage.structure.element import Element
from sage.combinat.free_module import CombinatorialFreeModule

# TODO: doc and tests for all of those
class SymbolicMatrix(SageObject):
def __init__(self, name, nrows, ncols, inverted = False, transposed = False):
#Element.__init__(self, parent)
self._name = name
self._nrows = nrows
self._ncols = ncols
self._inverted = inverted
self._transposed = transposed

def _repr_(self):
result = self._name
if self._inverted:
result += "^-1"
if self._transposed:
result += "^t"
return result

def transpose(self):
result = copy(self)
result._transposed = not self._transposed
(result._nrows, result._ncols) = (self._ncols, self._nrows)
return result

def __invert__(self):
assert self._nrows == self._ncols, "Can't inverse non square matrix"
result = copy(self)
result._inverted = not self._inverted
return result

class SymbolicMatrixAlgebra(CombinatorialFreeModule):
r"""

EXAMPLES::

sage: Alg = SymbolicMatrixAlgebra(QQ)

sage: A = Alg.matrix("A",3,2)
sage: B = Alg.matrix("B",2,3)
sage: C = Alg.matrix("C",2,3)
sage: D = Alg.matrix("D",3,3)

Example 1: Adding/Multiplying matrices of correct size::

sage: A * (B + C)
A B + A C

Example 2: Transposing a sum of matrices::

sage: (B + C).transpose()
C^t + B^t

Example 3: Transposing a product of matrices::

sage: (A * B).transpose()
B^t A^t

Example 4: Inverting a product of matrices::

sage: (A * B)^-1
B^-1 A^-1

Example 5: Multiplying by its inverse::

sage: D * D^-1 # todo: not implemented
I

TODO: decide on the best output; do we want to be able to
copy-paste back? do we prefer short notations?

.. warnings::

The identity does not know it is size, so the following should
complain, but does not::

sage: D * D^-1 * A

TODO: describe all the abuses
"""

def __init__(self, R):
"""
EXAMPLES::

sage: A = AlgebrasWithBasis(QQ).example(); A
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: TestSuite(A).run()

"""
CombinatorialFreeModule.__init__(self, R, Words(), category = AlgebrasWithBasis(R))

def matrix(self, name, nrows, ncols):
""" TODO: doctest"""
return self.monomial(Word([SymbolicMatrix(name, nrows, ncols)]))

def _repr_(self):
"""
EXAMPLES::

sage: SymbolicMatrixAlgebra(QQ)
The symbolic matrix algebra over Rational Field
"""
return "The symbolic matrix algebra over %s"%(self.base_ring())

@cached_method
def one_basis(self):
"""
Returns the empty word, which index the one of this algebra,
as per :meth:AlgebrasWithBasis.ParentMethods.one_basis.

EXAMPLES::

sage: Alg = SymbolicMatrixAlgebra(QQ)
sage: Alg.one_basis()
word:
sage: A.one()
I
"""
return self.basis().keys()([])

def product_on_basis(self, w1, w2):
r"""
Product of basis elements, as per :meth:AlgebrasWithBasis.ParentMethods.product_on_basis ...
more

1
That's an interesting thread, which I had forgotten. Did it ever turn into a ticket?  ( 2011-04-12 16:46:18 -0500 )edit
@kcrisman - I had a quick look, but couldn't find anything.  ( 2011-04-13 15:20:01 -0500 )edit
Then open one and post the code :)  ( 2011-04-14 03:45:30 -0500 )edit
@kcrisman: Remind me again in a couple of months after I've written my thesis - then I'll have a go at completing and posting the code...  ( 2011-04-15 13:32:21 -0500 )edit

answered 2011-04-12 06:04:35 -0500

I'm not sure if there's a way to define a symbolic matrix the way you describe above. However, one could create a matrix populated only by distinct symbolic variables. Here's a quick, though supposedly not quickest, way to do so:

sage: N = 3
sage: s = join(['a_%d%d' %(i,j) for (i,j) in CartesianProduct(range(N),range(N))])
sage: a = var(s)
sage: A = matrix(SR,N,N,a)
sage: print A[1,2]
a_12

The second line just creates a string with the variable names $a_{ij}$ for $i,j \in \mathbb{Z}_n$ and the third parses the string and creates a list of the symbolic variables. I can now create another matrix $B$ and multiply them:

sage: s = join(['b_%d%d' %(i,j) for (i,j) in CartesianProduct(range(N),range(N))])
sage: b = var(s)
sage: B = matrix(SR,N,N,b)
sage: C = A*B
sage: print C[1,2]
a_10*b_02 + a_11*b_12 + a_12*b_22

It's not completely automatic, and perhaps not pretty, but I hope it helps.

more

This isn't a very "beautiful" method, but I can't think of anything better. :-) Anybody know of a cleaner approach? By the way, I would put an underscore between the indices, so the variables would read: "a_1_0", "b_3_5", and "c_23_829" (just an example). And would this method work for sparse matrices?  ( 2011-04-12 06:46:33 -0500 )edit
I agree that it's not pretty, but even if Sage had the ability to define these "purely symbolic" matrices one would have to call the elements _something_. Perhaps this sort fo thing would happen in the __init__() method. Also, I use the notation "a_ij" instead of "a_i_j" because when you execute "show(a_ij)", or enable typesetting in the Sage Notebook, it looks prettier. :)  ( 2011-04-12 07:29:03 -0500 )edit
Ignoring all else, I would like to reference and name the elements consistently: a[1,2] being printed exactly as "a[1,2]". Then again, I can't ignore the facts. :-)  ( 2011-04-12 08:17:42 -0500 )edit
@cswiercz: This is basically how you have to do things in Mathematica - you then check results for a range of N` and hope that that's enough. I was sure that sympy used to have an abstract matrix module... but I couldn't find it when I looked just now.  ( 2011-04-12 13:30:23 -0500 )edit
You say it's not the quickest way to do this (*this* being define lots of symbolic variables and populate a matrix with them), so what is the quickest way?  ( 2011-04-19 05:34:06 -0500 )edit

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