Principal vectors of a nonlinear finite-dimensional eigenvalue problem (Q2629999)

The authors deal with a nonlinear eigenvalue problem analytic with respect to its spectral parameter, in a finite-dimensional linear space. A new approach to the notion of principal vectors is introduced. More precisely, the authors provide the definition of principal vectors of grade greater than one for the considered eigenvalue problem exploiting the property that principal vectors are equal to the derivatives of eigenvectors with respect to the spectral parameter under small perturbations of the matrix entries. It is shown that the concept of principal vectors given in this paper is identical to the conventional definition in the case of a linear eigenvalue problem. It is also proved that the multiplicity of an eigenvalue is equal to the maximum number of linearly independent eigenvectors combined with principal (associated) vectors in the corresponding chains. According to this approach, a numerical method to calculate principal vectors is provided.