{\mathbf{q}}_{m \alpha} \times {\mathbf{H}}_{0m \alpha}$$ The (local) material tensor is represented, in general, by a $6 \times 6$ matrix. Since the number of columns in matrix $\Psi_{EH}$ is typically greater than the number of rows, the matrix equation for the material tensor is solved in the least squares sense: $$\label{eqn:M-eq-DB-over-EH}
{\mathcal{M}}\,=\, \Psi_{DB} \Psi_{EH}^+;    
~~~~~ \delta_{\mathrm{l.s.}} \,=\, \| \Psi_{DB} - {\mathcal{M}}\Psi_{EH} \|_2$$ where $\Psi_{EH}^+$ is the Moore-Penrose pseudoinverse of $\Psi_{EH}$, and $\delta_{\mathrm{l.s.}}$ is the associated least-squares error.

![Example A of a layered medium from [@Markel-Tsukerman-PRB2013; @Tsukerman-Markel14]. The real part of $R$ (left) and $T$ (right) vs. the sine of the angle of incidence; non-asymptotic and nonlocal homogenization. The lattice cell contains three layers of widths $a/4$, $a/2$ and $a/4$, with scalar permittivities $\epsilon_1$, $\epsilon_2$, and $\epsilon_1$, respectively. ($\epsilon_1 = 4 + 0.1i$ and $\epsilon_2 = 1$.) Fine-level basis: $2n_{\mathrm{dir}}$ Bloch modes traveling at $n_{\mathrm{dir}} = 7$ different angles in $(-\pi/2, \pi/2)$; $n_{\mathrm{dir}} = 7.$ The kernel width parameter $\tau_0 = a$. The reflection and transmission coefficients from nonlocal homogenization are visually indistinguishable from the exact ones. The nonlocal procedure includes two additional DoF: the convolution integrals of the tangential components of the electric and magnetic fields.[]{data-label="fig:Re-RT-vs-angle-ExA-PRB2013"}](Re_R_vs_angle_Ex_A_PRB2013 "fig:"){width="0.85\linewidth"} ![Example A of a 