In this paper we apply the $\U$-Lagrangian theory to the maximum eigenvalue function $\La$ and to its precomposition with affine matrix-valued mappings. We first give geometrical interpretations of the $\U$-objects that we introduce. We also show that the $\U$-Lagrangian of $\La$ has a Hessian which can be explicitly computed; the second-order development of the $\U$-Lagrangian provides a second-order development of $\La$ along a characteristic smooth manifold: the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity. The same results can be obtained when we precompose $\La$ with an affine matrix-valued mapping A, provided that this mapping satisfies a regularity condition ({\em transversality condition}). We show that the Hessian of the $\U$-Lagrangian of $\La \circ A$ coincides with the reduced Hessian encountered in sequential quadratic programming. Finally, we use the $\U$-Lagrangian to derive second-order algorithms for minimizing $\La \circ A$.