If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite trivially, since $x^TA^TAx = \|Ax\|^2$, so I guess it will not be positive-definite only if the random matrix $A$ is rank-deficient, which should almost never happen.

But from numerical stability point of view, things *might* look different, because the matrix $A^TA$ could be close to a p.s.d. non-p.d. matrix with some probability.

So: is it generally safe to apply Cholesky decomposition to such a matrix, or is there a nontrivial chance of that leading to numerical instability? (I.e. without stronger conditions which would force the matrix away from the problematic boundary of p.d. matrices with non-p.d. matrices?)