How can I simplify the result of a symbolic expression with it's coefficients to an expression that only uses the variable names?

Hello,

I would like to use the symbolic toolbox to validate some equations and I was wondering if it is possible to simplify a result so that it does not show all coefficients but only the defined symbol names. The example only shows a simplified version of the problem, the actual equations are more complex.

Example:

a = sym("a", [3 3])
b = sym("b", [3 3])
c = sym("c", [3 1])

result = a*b*c

Result:

 c1*(a1_1*b1_1 + a1_2*b2_1 + a1_3*b3_1) + c2*(a1_1*b1_2 + a1_2*b2_2 + a1_3*b3_2) + c3*(a1_1*b1_3 + a1_2*b2_3 + a1_3*b3_3)
 c1*(a2_1*b1_1 + a2_2*b2_1 + a2_3*b3_1) + c2*(a2_1*b1_2 + a2_2*b2_2 + a2_3*b3_2) + c3*(a2_1*b1_3 + a2_2*b2_3 + a2_3*b3_3)
 c1*(a3_1*b1_1 + a3_2*b2_1 + a3_3*b3_1) + c2*(a3_1*b1_2 + a3_2*b2_2 + a3_3*b3_2) + c3*(a3_1*b1_3 + a3_2*b2_3 + a3_3*b3_3)

But what I actually would like to see is:

a*b*c

Answer 1

[Walter Roberson]

There is no operation for that in the Symbolic Toolbox. Internally, the symbolic engine MuPAD has no provision for the generalized notion of a "matrix" that might be matrix-multiplied as a whole: the symbolic engine only knows about arrays that are populated with something and whose individual elements are acted on.

[Walter Roberson]

A further note:

if you use

syms a b c
d = a*b*c

expecting that a, b, and c will "stand in" for matrices, and that later you will be able to subs() in matrices and have the proper operations carried out -- then you are going to be disappointed.

Because the symbolic engine has no internal knowledge of "matrices" as distinct objects, any time it sees individual symbols "*" together, it assumes that they are scalars and that commutative multiplication will be used.

>> syms a b c
>> a*b*a
ans =
a^2*b
>> a*b*inv(a)
ans =
b

This makes the symbolic engine useless for having named variables standing in for matrices with matrix algebra rules expected.

It also implies that if you move to the level of individual matrix components and calculate a formula, that you cannot unambiguously reconstruct it in terms of matrix operations. You cannot take cues from the order of variables and expressions such as a1_1*b1_1 + a1_2*b2_1 + a1_3*b3_1 because the symbolic engine will rewrite those according to undocumented rules.

>> - a1_1*b1_1 - a1_2*b2_1 + a1_3*b3_1
ans =
a1_3*b3_1 - a1_2*b2_1 - a1_1*b1_1

It tends to prefer positive leading terms, but there are circumstances under which it will prefer negative leading terms (especially if there is an external negative multiplier, leading to potential representations where the apparent negative leading term was stored internally as a positive leading term but brought out as negative for display purposes...)

[Chris B]

This is an important information, thank you. I did not know that the symbolic toolbox does not respect the properties of matrices. Is there a way to define that matrix multiplications are non-commutative? Or is there another toolbox that is more suitable for this task?

[Walter Roberson]

I think it just might be possible to go into the symbolic engine (MuPAD) and tell it to define a new DOMAIN and then fill in function "slots" for each desired operation. It might be a bit tricky to have objects of that DOMAIN cross back and forth between MATLAB and the symbolic engine, but I think something usable might be possible.

When I looked at this a few months ago, I seem to recall that I did have trouble with the print slots (character representation for the MATLAB side), but it has been long enough that I no longer recall exactly what I did.