Factorizing a matrix into a matrix and its transpose Let $W\in \mathbb{R}^{n \times n}$ be a positive semi-definite matrix. 

Then, what are some well-known factorization methods that guarantee $W=A^T A$, with the conditions being that 

\begin{align}
1.&amp; \ \ \ \ \ A \in \mathbb{R}^{n \times n}, \\
2.&amp; \ \ \ \ \ \text{$A$ has the same rank as $W$}?
\end{align}
It's not possible to factor a general positive semidefinite matrix that way. 

$A^TA$ is automatically symmetric. But there are positive semidefinite matrices that are not symmetric. Like $W=\begin{bmatrix}1&amp;3\newline-1&amp;1\end{bmatrix}$, which has $\vec{v}^tW\vec{v}=(v_1+v_2)^2\geq0$ for all $\vec{v}$.

Did you intend to ask about factoring a symmetric matrix? Or a Hermitian positive semidefinite matrix?
I don't know such a decomposition for all p.s.d. matrices $W$. But take a look at Matrix decomposition especially Cholesky decomposition.