Can You Determine a Matrix M from Quadratic Forms?

[Chuck37]

Not sure what to call this problem, but say I have a bunch of equations like this:

v1'*M*v1 = c1
v2'*M*v2 = c2
...

M is an unknown square matrix and is the same in every equation. v is a vector and c is a scalar, these are different in each equation. I can have as many of these as I need. Is there a clever way to solve for M? I wrote out the 2x2 case and can solve it if I take apart M, but I'd like a more elegant extensible solution if one exists.

I hope my question makes sense. Thanks.

 

[fresh_42]

If you can choose the $v_i$ at your will, then you can take $v_i=(0,\ldots,0,1,0,\ldots,0)=(\delta_{ij})_j$ which gives you the diagonal. But there is no easy shortcut. If you only have these equations, then you need $n^2$ of them to solve your linear equation system
$$
c^{(m)} = \sum_{i,j} v_i^{(m)}X_{ij}v_j^{(m)} \text{ where } M=(X_{ij})_{i,j}
$$
Of course, with specific choices of the $v_i$ you can simplify it, otherwise you cannot.