Characteristic values of the Jacobian matrix and global invertibility (Q2724115)

Let \(F = (F_1, \dots, F_n) : \mathbb R^n \to \mathbb R^n\) be a \(\mathcal{C}^2\)-map and \(J(F) = (\tfrac{\partial F_i}{\partial X_j})\) its Jacobian matrix. Assume that \(\det J(F) (x) \not= 0\) for all \(x \in \mathbb R^n\). This implies that \(F\) is a local diffeomorphism. The global invertibility is related to bounds of the singular values of \(J(F) (x)\) [cf. \textit{J. Hadamard}: ``Sur les correspondences ponctuelles'', Oeuvres, Editions CNRS, Paris, 383-384 (1968; Zbl 0168.24101)]. The so-called characteristic matrix values, as singular values, eigenvalues, pivots arising from Gaußian elimination of the Jacobian matrix and its inverse are considered and related to the global invertibility of the map \(F\).