Irrationality of linear combinations of eigenvectors (Q1917760)

Let \(A\) be a rational nonsingular \(n \times n\) matrix whose characteristic polynomial \(\chi (A)\) is irreducible over \(\mathbb{Q}\). Let \(\Omega\) be the set of the \(n\) distinct eigenvalues of \(A\), and let \(B\) be an \(n \times n\) matrix over \(\mathbb{C}\) whose rows are eigenvectors of \(A\) for the different eigenvalues. Let \(\Gamma\) denote the Galois group of the splitting field of \(\chi(A)\) over \(\mathbb{Q}\). Then the main Theorem is: If \(\Gamma\) acts transitively on the set \(\Omega_q\) of all subsets of size \(q\) in \(\Omega\) for each \(q \leq r\), then all \(r \times r\) minors of \(B\) are nonsingular. Two consequences are: (1) If \(\Gamma\) acts as the symmetric or alternating group on \(\Omega\), then any set of \(n\) vectors taken from the set of rows of \(B\) or the set of standard basis vectors of \(\mathbb{C}^n\) is linearly independent; and (2) no eigenvector of \(A\) can be written as a \(\mathbb{C}\)-linear combination of \(n - 1\) vectors from \(\mathbb{Q}^n\).