is it possible to write Gram matrix by a system of 3 vectors

Question

Given a Gram matrix G 3x3 = {{25,-14,-9},{-14,12,14},{-9,14,25}} is there such a system of 3 vectors in R^3 whose gram matrix is ​​equal to G. I have a stupid idea to write down 6 equations (from Gram matrix rule), but I don't know what to do next. IS there any algorithm?

Answer 1

You first need to check whether your matrix is symmetric positive-semidefinite. Then, you can use Cholesky decomposition to compute the vectors.

Answer 2

The determinant is 256 = 16^2, so there is a chance this can be done with just integers.

$$ \left( \begin{array}{rrr} -4 & - 3 & 0 \\ 2 & 2 & 2 \\ 0 & 3 & 4 \\ \end{array} \right) \left( \begin{array}{rrr} -4 & 2 & 0 \\ -3 & 2 & 3 \\ 0 & 2 & 4 \\ \end{array} \right) = \left( \begin{array}{rrr} 25 & - 14 & - 9 \\ - 14 & 12 & 14 \\ - 9 & 14 & 25 \\ \end{array} \right) $$