Chebotar\"ev's nonvanishing minors for eigenvectors of random matrices and graphs

For a matrix $\mathbf{M} \in \mathbb{K}^{n \times n}$ we establish a condition on the Galois group of the characteristic polynomial $\varphi_\mathbf{M}$ that induces nonvanishing of the minors of the eigenvector matrix of $\mathbf{M}$. For $\mathbb{K}=\mathbb{Z}$ recent results by Eberhard show that, conditionally on the extended Riemann hypothesis, this condition is satisfied with high probability and hence with high probability the minors of eigenvector matrices of random integer matrices are nonzero. For random graphs this yields a novel uncertainty principle, related to Chebotar\"ev's theorem on the roots of unity and results from Tao and Meshulam. We also show the application in graph signal processing and the connection to the rank of the walk matrix.