First define a function $P$ mapping matrix into matrix of dimension $m$ by $m$, given $V$ and $L$ $$P(A)=V^T(L\otimes A)(L\otimes A)^TV$$ where $L$ is $N$ by $N$ and $V = [v_1,v_2,\ldots,v_m]$, {$v_i$} is orthonormal in $\mathbf{R}^{Nn}$ and $A$ is a $n$ by $d$ matrix with constraints $d<n$ and full column rank, is there any way to find eigenvector of $P$ (in terms of something related to $A$) ?
My thought is to use SVD of $L$ and $A$ $$P(A)=V^T(U_L\otimes U_A)(\Sigma_L\otimes \Sigma_A)(\Sigma_L\otimes \Sigma_A)^T(U_L\otimes U_A)V = V^TU_{LA}D_{LA}U_{LA}^TV =SD_{LA}S^T$$
where I denote $(U_L\otimes U_A)$ by $U_{LA}$ and $D_L \otimes D_A$ by $D_{LA}$ ($D_L = \Sigma_L\Sigma_L^T$, $D_A = \Sigma_A\Sigma_A^T$ ) and $S = V^TU_{LA}$
Then we have a form look like eigen-decomposition ($SS^T = I_m$) but the dimension of $S$ is $m$ by $Nd$ (if we apply thin or economy SVD) not $m$ by $m$, I have no idea about future steps. Any help would be appreciated.