Simplifying a matrix expression with a MatrixSymbol in Python

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I am using SymPy for symbolic matrix calculations, however, certain statements are very big. It seems there is a way to simplify them further. I have used simplify() but I have not been successful in getting what I want.

For example the image below is a matrix, which has been obtained as result of execution of long list of previous matrix calculations. The output of series of matrix calculations, which requires further simplification

The final statement has two additions and one matrix multiplication. I wonder if there is any way to also execute the matrix multiplication on the right so we can simply end up with 3 matrix summations?

I understand this can be done by doing certain algebraic manipulations by hand but I am more interested in a command to execute, such that the command takes the whole statement as the input and does all the simplifications including any multiplications and additions and outputs what I need. This should all be done using sympy. In other words, if an addition or a multiplication can be done then I want it done and not being left undone.

This is MCVE, which mimics my problem

from sympy import *
init_printing()
J_22 = MatrixSymbol('J_22', 3, 3)
COV_b=Matrix([[2,1,1],[1,2,1],[1,1,2]])
(COV_b+J_22)*COV_b

The output of this code is

The output of the MCVE

However, I would like to have this as the output

The desired output

I understand that in this simple example I could simply fix the problem by the following code

from sympy import *
init_printing()
J_22 = MatrixSymbol('J_22', 3, 3)
COV_b=Matrix([[2,1,1],[1,2,1],[1,1,2]])
(COV_b*COV_b+J_22*COV_b)

However, this is just a simple example, in the actual problem this cannot be seen before the output is generated. Hence, I would like to be able to use a command, which takes the output of the first provided code as input and outputs the desired output.

UPDATE:@WelcometoStackOverflow provided a function, which simplifies things a lot but still leaves the matrix addition undone.

from sympy import *
init_printing()
J_22 = MatrixSymbol('J_22', 3, 3)
COV_b=Matrix([[2,1,1],[1,2,1],[1,1,2]])
T=(COV_b+J_22)*COV_b+COV_b
def expand_matmul(expr):
    import itertools
    for a in preorder_traversal(expr):
        if isinstance(a, MatMul):
            terms = [f.args if isinstance(f, MatAdd) else [f] for f in a.args]
            expanded = Add(*[MatMul(*t) for t in itertools.product(*terms)])
            if a != expanded:
                expr = expr.xreplace({a: expanded})
                return expand_matmul(expr)
    return expr
expand_matmul(T)

The output is

enter image description here[4]

with the summation between the first two matrices still not executed.

sympy python
1 Answers

This is a known and old issue with SymPy expressions: Can't expand matrix expression. Matrix Expressions module is useful, but is not the most actively maintained in SymPy. I put together a function to expand such things.

def expand_matmul(expr):
    import itertools
    for a in preorder_traversal(expr):
        if isinstance(a, MatMul) and any(isinstance(f, MatAdd) for f in a.args):
            terms = [f.args if isinstance(f, MatAdd) else [f] for f in a.args]
            expanded = MatAdd(*[MatMul(*t) for t in itertools.product(*terms)])
            if a != expanded:
                expr = expr.xreplace({a: expanded})
                return expand_matmul(expr)
    return expr

The function walks the expression tree from the highest levels, looking for an opportunity to expand MatMul. The returned expression may benefit from doit method invocation to perform any undone multiplication from explicit matrices, as in the example below. 

J_22 = MatrixSymbol('J_22', 3, 3)
COV_b=Matrix([[2,1,1],[1,2,1],[1,1,2]])
T=(COV_b+J_22)*COV_b+COV_b  
pprint(expand_matmul(T).doit())

prints

⎡8  6  6⎤       ⎡2  1  1⎤
⎢       ⎥       ⎢       ⎥
⎢6  8  6⎥ + J₂₂⋅⎢1  2  1⎥
⎢       ⎥       ⎢       ⎥
⎣6  6  8⎦       ⎣1  1  2⎦
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