I have a Laplacian matrix of a graph with $N$ nodes named $L_1$. We can diagonalize the matrix and write it as follow:
$L_1 = U\Lambda U^T = \sum_k \lambda_ku_ku_k^T = \sum_k \lambda_kB_{kk}$
I have another Laplacian matrix of another graph with $N$ nodes named $L_2$. I want to write $L_2$ based on the eigenvectors of $L_1$ (I mean $U$).
My Attempt: For this purpose, I thought that I can define the basis for 2D matrices as $B_{ij} = u_iu_j^T$. So, I want to find $\alpha_{ij}$ for the following expression:
$L_2 = \sum_i\sum_j \alpha_{ij}B_{ij}$
But I suddenly found out that the basis that I've defined are not orthogonal to each other. I think that for orthogonality, the following conditions should be satisfied:
$B_{ij}^TB_{pq} = 0 \; \; \forall i,j,p,q , i\neq j, p\neq q$
$B_{ij}^TB_{ij} = I$
But, $B_{ij}^TB_{iq} = u_ju_i^Tu_iu_q^T = u_ju_q^T$ which is not identity. Is there any other way, I can write Laplacian of one