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

Question

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.

Answer 1

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 $A^* A = A^2 = I$.