I was trying the calculate the eigenvectors of the adjacency matrix of a bipartite graph by utilizing the bi-adjacency matrix given by $B \in \mathbb{R}^{m \times n}$. The adjacency matrix of the whole graph can be defined as $$A = \begin{bmatrix}0 & B \\ B^\top & 0 \end{bmatrix}.$$
If $B$ has SVD given by $B = U\Sigma V^\top$, then the eigendecomposition of $A$ is given by $$A = \begin{bmatrix}\bar U & \bar U \\ \bar V & -\bar V\end{bmatrix} \begin{bmatrix} +\Sigma & \\ & -\Sigma\end{bmatrix} \begin{bmatrix} \bar U & \bar U \\ \bar V & -\bar V\end{bmatrix}^\top$$ where $\bar{U} = U/\sqrt{2}$.
My main aim to calculate the eigenvalues by only using the bi-adjacency matrix $B$ since $A$ might be too big. I was trying to verify this by using libraries like pytorch. First I compute $U,V$ using SVD and then I am trying to compare them with the eigenvectors of $A$ by doing eigendecomposition of $A$. However, the vectors do not match. Is there anything I am missing?
Image source: https://arxiv.org/abs/1412.6073