I am currently studying how to transform ODEs to SDEs and stumbled upon the paper "Construction of Equivalent Stochastic Differential Equations. In the paper, there is an example using the second modeling procedure of transforming ODEs to SDEs.
Given that $V = GG^{T}$, where $V$ is a known $3 \times 3$ symmetric positive semidefinite matrix and $G$ is $3 \times 4$.
I have read about Singular Value Decomposition (SVD) where $A = U\Sigma V^{T}$, and tried to factor $G$ as $G = P \Sigma Q^{T}$, where $P$ is the eigenvectors of $GG^{T}$ and $\Sigma$ is the eigenvalues. However, $Q^{T}$ is still unknown (is it?).
I also read about Cholesky decomposition, where $A = LL^{T}$, and tried to factor $V$ as $V = GG^{T}$. Yes, I can get G but G is $3 \times 3$. I need a G that is $3 \times 4$.
What am I missing? Am I doing it wrong?