Appendix 4—figure 2. The $3\times 3$-matrix invariant $I_2=\frac{1}{2}\left(\mathrm{tr}[{(M^J)}^2]-{(\mathrm{tr}[M^J])}^2\right)$ sampled for pairs of positions $\mathbf{r}$, ${\mathbf{r}}^{\prime }$ vanishes to machine precision for the dynamical matrix $M$ learned on the zebrafish data.

This invariant can be expressed in terms of matrix eigenvalues as $I_2={\lambda }_1{\lambda }_2+{\lambda }_1{\lambda }_3+{\lambda }_2{\lambda }_3$. Additionally, ${\lambda }_1{\lambda }_2{\lambda }_3=\det (M^J)=0$ (see discussion below Equation 53), which implies only one eigenvalue is non-zero. Evaluating I₂ for the kernel matrix $M^J$ encoded by the theoretical (see Equation 13b and Equation 13c) and inferred (see Figure 3C, main text) dynamical matrix $M$ of the ABP dynamics yields similar results.