Decompose Gram Matrix
Suppose I have a real matrix X of size m*n. Let Y=X'X be the gram matrix. If someone knows Y and the size of X (m*n) only. Is he possibly find out what X is?
I know he can do the cholesky decomposition to Y to get a triangle matrix Z, such that Y = Z'Z. Does that mean there're infinite solutions?
If consider every elements in X as a variable. Then we can think of X'X = Y as a system of multivariate quadratic equations, where the number of variables is m*n and the number of equations is n. Does this system has infinite solutions? If it does, how to prove?
Thank you very much.