How to extract the "coefficients" of a symbolic matrix?

Given a symbolic square matrix y (Matlab code: syms y [n n] matrix;), for a symbolic matrix of the form y+Ay+yB+CyD+EyF+... (where A, B, C, D, E, F... are numerical matrix), how to separate the numerical matrices such as matrices A, B, C, D, E, F... through Matlab?

Example:

syms y [3 3] matrix;
A=randi(2,3,3);
B=randi(2,3,3);
C=randi(2,3,3);
D=randi(2,3,3);
x=y+A*y+y*B+C*y*D

run the code and then I got:

y*symmatrix([2,1,1;2,1,1;1,1,1]) + symmatrix([2,2,1;1,2,2;2,2,2])*y + symmatrix([1,1,2;1,1,1;1,2,2])*y*symmatrix([2,2,2;2,2,1;1,1,1]) + y

Then my question is: give the x=f(y) like above,how do I get all the matrices where y is multiplied by left and right?

I looked up the functions in the Symbolic Math Toolbox and Googled related questions, unfortunately all I could find was about the coefficients of symbolic polynomials or other expressions, and the "coefficients" of symbolic matrices were hard to find relevant answers.

I would appreciate it if you could help me.

[Paul]

I thought we might be able to get there with children, but I was wrong

syms y [3 3] matrix;
A=randi(2,3,3);
B=randi(2,3,3);
C=randi(2,3,3);
D=randi(2,3,3);
x=y+A*y+y*B+C*y*D

x = 

children(x)

Incorrect number or types of inputs or outputs for function 'children'.

Answer 1

[Steven Lord]

Let's simplify the problem a bit and see if we can solve this for the expression z = y + A*y + y*B. What happens if either A or B commute with y? Then we could write this as (if A commutes with y) z = y + y*A + y*B which simplifies to z = y*(I+A+B). How would you determine unique matrices A and B that generate your result z? The answer is you don't, not without more information or making an arbitrary choice (like A is the identity I which is guaranteed to commute with y.)

For the 1-D case see this post on Cleve Moler's blog. In Cleve's impossible problem we're trying to find A and B that satisfy 3 = A*(1/2)+(1/2)*B. Obviously (1/2) commutes with both A and B so "the" solution is not unique.

So do you have more information about or constraints on your A, B, C, D, etc. matrices?

[TRI]

Unfortunately, the problem I need to deal with with matlab is that X is just a general symbolic matrix, generally non-commutative, with no exact numeric characteristics