How to find unknown linear operator?

I have a square matrix A that acts on a some vector x such that A*x = y. Matrix A is heavily constrained with only a few unknowns. x and y are known exactly. How do I solve for A?

For example say:

sym a b c d
A = [a 0 0 b;
     c d 0 0;
     0 0 0 0;
     0 0 0 0]
x = % some known vector
y = % some known vector

How do we find A now?

[sahil kommalapati]

This is a method which come to mind:

syms a b c d
A = [a 0 0 b;
     c d 0 0;
     0 0 0 0;
     0 0 0 0];
 
x = zeros(4,1); %your known vecotor here
y = zeros(4,1); %and here!
eq1 = A*x ==y;

k = solve(eq1, [a,b,c,d]);

sol = [k.a, k.b, k.c, k.d]

[David Goodmanson]

Hello Tsuyoshi,

Well, with your example for A, you are certainly not going to do it for arbitrary x and y, because the form of A means that y(3)=y(4)=0. So any solution with either y(3) or y(4) nonzero is impossible. (It's a consequence of A being singular).

But let's say that y(3)=y(4)=0. The two equations that are left are

a*x(1) + b*x(4) = y(1)
c*x(1) + d*x(2) = y(2)

which is one eqn. for two unknowns a,b

and one eqn. for two unknowns c,d

so you can get solutions, but you can't solve for any unknown uniquely.