# Simultaneous diagonalization and SVD of 4th order tensor

Given a 4th order tensor $L_{ijkl}$, how can I find four unit vectors $\mathbf x$, $\mathbf y$, $\mathbf z$ and $\mathbf w$ such that:

1. $\mathbf{w}$ and $\mathbf z$ are the left and right singular vectors corresponding to the largest singular value of $$L_{ijkl}\,(x_ix_j^* + y_iy_j^*)$$

2. $\mathbf x$ and $\mathbf y$ are the eigenvectors corresponding to the two largest eigenvalues of $$L_{ijkl}\,z_kw_l^*$$ (With the optimal $\mathbf z$ and $\mathbf w$ this is a positive matrix).

I looked in the literature of tensor decompositions, but nothing I found helped me so far.