How to undo a matrix-vector multiplication

Question

I have an iterative algorithm that computes a matrix-vector multiplication such as:

$$ b = Av $$

I know the vector b (which is the result of the algorithm) and the vector v. Is there a way to get the matrix A?

Answer 1

The first column of $A$ is $A.(1,0,0,\ldots,0)$, the second column is $A.(0,1,0,\ldots,0)$ and so on.

Answer 2

There exists an infinity of solutions to the problem $$b=Av$$ Where $b$ and $v$ are vectors

The problem can be rewritten as $$b^T = v^T A^T $$

The minimum (least square) solution is given by the Moore-Penrose pseudo-inverse: $$A_0^T = (v^T)^+ b^T = v\,(v^T v)^{-1}\, b^T$$

Which gives $$A_0 = \frac{1}{|v|^2} b\,v^T$$

Note that $A_0$ is of rank $1$.

The other solutions are given by $$A = A_0 + B = \frac{1}{|v|^2} b\,v^T + B $$ For any matrix $B$ such that $B\,v = 0$