TITLE: Are all symmetric matrices with positive eigenvalues a product of a matrix and its transpose?
QUESTION [6 upvotes]: Given an $m \times n$ matrix $A$, it's easy to show that the matrix products $B = A^TA$ and $C = AA^T$ are both symmetric. I was wondering if any symmetric matrix with positive eigenvalues could be expressed as the product of some matrix and its transpose? 
Note: It's obvious that if the eigenvalues are not all positive then this cannot be the case, as $A^TA$ and $AA^T$ both have to be positive semidefinite. However this is all I can think of regarding this problem.

REPLY [9 votes]: A direct answer is that, if a matrix is symmetric and has $>0$ eigenvalues, thus is a symmetric positive definite matrix, it has a Cholesky decomposition $A=C^TC$ (https://en.wikipedia.org/wiki/Cholesky_decomposition), with an upper triangular $C$.