# Can we solve a matrix equation when the vectors are given and the matrix is variable?

Usually, a matrix equation means that $$Ax = b$$ when A and b are given and x is the variable we want to know.

However, when x and b are given and we want to know the value of the matrix, is it acheivable?

To provide context, I'm studying quantum computing and I tried to figure out if a certain state vector can be derived from $$|00...0 \rangle$$ with unitary transformations. So you can assume $$A$$ is unitary and $$b$$ is $$|00...0 \rangle$$.

Now I'm almost brute-forcing to solve this kind of equation, so even though there is no general algorithm for it, just giving some tips and tricks would also be very helpful.

If $$A$$ is unitary then $$x$$ and $$b$$ must have the same magnitude; otherwise it is impossible.
If $$x$$ and $$b$$ have the same magnitude, then yes there exist many unitary matrices that satisfy the equation. One such example is the reflection
$$A = I - 2 \frac{(x-b)(x-b)^*}{(x-b)^*(x-b)}$$ where the $$*$$ indicates conjugate transpose. You can check that $$A$$ us unitary by verifying $$A^*=A$$ and $$A^* A = A^2 = I$$.