Summary: The eigenvalues of the fourth-order constitutive tangent modulus and the corresponding acoustic tensors are analyzed. Explicit expressions of the eigenvalues are made for the nonsymmetric tangent modulus tensor, and in the case of the deviatoric associative rule for the symmetric part of the tangent modulus and its acoustic tensor. In this context, a rate independent infinitesimal elastoplastic model is considered. The expressions of the plastic hardening modulus are summarized for the different local stability criteria (loss of second-order work positiveness, loss of ellipticity, and loss of strong ellipticity). The critical hardening modulus and orientation are discussed in detail in the case of loss of ellipticity and loss of strong ellipticity. This analysis is based on the geometric method and linear, isotropic elasticity and deviatoric associative flow rule. In particular, the critical orientation for the loss of strong ellipticity and the classical shear band localization are compared.