PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

SUBDIFFERENTIAL OF THE LARGEST EIGENVALUE OF A SYMMETRICAL MATRIX APPLICATION OF DIRECT PROJECTION METHODS

In many physical circumstances, for example, in studying the linear vibrations of a mechanical or acoustical system, a key tool is to determine numerically the components of the eigenvectors associated with the largest eigenvalue of a symmetrical matrix with real coefficients. To find out the largest eigenvalue λ1(S) of such a symmetrical n × n matrix S, the well-known Rayleigh's method consists in maximizing the quotient (VTSV)/(VTV) among all the nonvanishing vectors V of ℝn. When the eigenvalue λ1(S) is simple, the maximum is attained for vectors V colinear to a unit eigenvector N, and the function λ1 is differentiable with the projector NNT over the direction N as a gradient. When the largest eigenvalue is not simple, the function λ1 is no longer differentiable; it remains convex, but the subdifferential ∂λ1(S) is not reduced to a single gradient. This paper is devoted to determine the subgradients therein ∂λ1(S) by direct methods that do not require the preliminary determination of λ1(S).