I have a Laplacian matrix ($L$), which is positive semi-definit. Then I have this matrix

$$\Delta=\begin{bmatrix} \delta_{11} & \ldots & \delta_{1n} \\ \vdots & \ddots & \vdots \\ \delta_{n1} & \ldots & \delta_{nn} \end{bmatrix}$$ in which $\delta_{ij}=e_ie_{j}^{T}$, and $e_i$ is a vector.

I need to form the following matrix, and analyze the eigenvalues.

$$\begin{bmatrix} l_{11}\delta_{11} & \ldots & l_{1n}\delta_{1n} \\ \vdots & \ddots & \vdots \\ l_{n1}\delta_{n1} & \ldots & l_{nn}\delta_{nn} \end{bmatrix}$$ where $l_{ij}$ is a scalar, but $\delta_{ij}$ is a matrix. This certainly looks like the Hadamard product, but the problem is that the dimensions of $L$ and $\Delta$ don't necessarily match.

Any ideas about the eigenvalues?