This work concerns the eigenvalues (more precisely, the spectral radius) of the Coxeter transformations \(C = C(A, \pi)\) for a generalized Cartan matrix \(A\) [for such matrices see \textit{V. Kac} ``Infinite dimensional Lie algebras'', 2nd ed. Cambridge Univ. Press (1985; Zbl 0574.17010); 3rd ed. (1990; Zbl 0716.17022)]. The following theorem is proved: Let \(A\) be a generalized Cartan matrix which is connected and neither of finite nor of affine type. Let \(C\) be a Coxeter transformation for \(A\). Then the spectral radius \(\rho (C) > 1\), and \(\rho (C)\) is an eigenvalue of multiplicity one, whereas any other eigenvalue \(\lambda\) of \(C\) satisfies \(| \lambda | < \rho (C)\).