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?)



Probability for a random positive-semidefinite matrix to not be positive-definite?
Not sure if this is what you are really after, but anyway: The paper by Rudelson and Vershynin that Igor Rivin [linked to](https://mathoverflow.net/a/149334/9652) contains lots of things which may be helpful for you. For example, for random $N\times n$ matrices ($N>n$) with iid Gaussian entries, there is Theorem 2.6 there which says that for the smallest singular value $s\_\text{min}(A)$ it holds that
$$\sqrt{N} - \sqrt{n}\leq \mathbb{E}s\_\text{min}(A).$$
Also, there is a the quantitative bound
$$\mathbb{P}(\sqrt{N}-\sqrt{n} - t \leq s\_\text{min}(A)) \geq 1 - 2\exp(-t^2/2).$$
The case of square (but not necessarily symmetric) matrices is more difficult. There is also a probability for a lower bound on the smallest singular value: For square (subgaussian) matrices, Theorem 3.2 says that for some $C>0$, $0<c<1$ and any $\epsilon>0$ it holds that
$$\mathbb{P}(s\_\text{min}(A)\leq \epsilon n^{-1/2})\leq C\epsilon + c^n.$$


