It may be easy for the expert.

Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$. My question is when this map is faithfully flat?

I known that for the faithfully part, (if already known it is flat) it is equivalent to the surjectivity of $\phi$. So the condition should at least be $m\geq n$.

I also known that when $m\geq 2n$, $\mathbb{C}[M_{n,m}] = \mathbb{C}[AA^T] \otimes H$ where $H$ is called Harmonics. So it is true for $m\geq 2n$.

I guess the map $\phi$ is faithfully flat exactly for $m\geq n$.

I also ask the similar questions for map $\phi\colon M_{n,r}\times M_{m,r}\to M_{n,m}$ by $(A,B)\mapsto AB^T$ and map $\phi\colon M_{2n, m} \to Alt_m$ by $A\mapsto A^TJA$ where $Alt_m$ is the space of antisymmetric matrices and $J$ is a symplectic form. The situations are similar, I guess the conditions are exactly the conditions for $\phi$ to be surjective, while I known the claim is true for a stronger condition.

All comments are welcome! Thank you very much!