Let $A$ be a complex $n\times n$ matrix. When does $XX^T = A$ have a solution?
My attempt:
Since $(XX^T)^T=XX^T$, $A$ has to be a symmetric matrix to have a solution.
After googling, I got to know about Cholesky Decomposition. According to this, if $A$ is real, then we can apply the decomposition for real matrices, and we have a solution.
Then, what about a complex case? I think Cholesky decomposition for complex matrices does not apply here because we have a transpose ($A^T$) instead of a conjugate transpose($A^*$).