  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.
