On complex eigenvalues and eigenvectors of the velocity gradient

According to Bertrand’s Theorem [J. Bertrand, Théorème relatif au mouvement le plus général d’un fluide, Comptes Rendus de l’Académie des Sciences de Paris, 66, (1867) 1127–1230], at any point in a continuum, at any time, there exists at least one infinitesimal material line element whose direction does not change, instantaneously. This direction is along an eigenvector of the velocity gradient tensor corresponding to a real eigenvalue. Corresponding to this eigenvalue there is also an infinitesimal material area element whose direction does not change, instantaneously. The Theorem does not address the question as to properties associated with complex eigenvalues of the velocity gradient. This note fills the gap. It is shown that corresponding to a complex eigenvalue there are two invariant ellipses, one associated with material line elements and the other with material areal elements.