Finding a vector from a matrix of element ratios

[enfield]

if I have square matrix A whose decimal elements $$a_{i,j} = \frac{v_i}{v_j}$$ how do I find (or approximate) the vector that would produce the matrix elements. what's a good way to do that?

i guess you can scale the vector however you want and it'll still produce the same matrix..

thanks.

 

[haruspex]

Consider what summing the matrix elements over i produces.