Generic change of the partial multiplicities of regular matrix pencils under low-rank perturbations Articles uri icon

publication date

  • January 2016

start page

  • 823

end page

  • 835

issue

  • 3

volume

  • 37

International Standard Serial Number (ISSN)

  • 0895-4798

Electronic International Standard Serial Number (EISSN)

  • 1095-7162

abstract

  • We describe the generic change of the partial multiplicities at a given eigenvalue lambda(0) of a regular matrix pencil A(0) + lambda A(1) under perturbations with low normal rank. More precisely, if the pencil A(0) + lambda A(1) has exactly g nonzero partial multiplicities at lambda(0), then for most perturbations B-0 + lambda B-1 with normal rank r < g the perturbed pencil A(0) + B-0 + lambda(A(1) + B-1) has exactly g - r nonzero partial multiplicities at lambda(0), which coincide with those obtained after removing the largest r partial multiplicities of the original pencil A(0) + A(1) at lambda(0). Though partial results on this problem had been previously obtained in the literature, its complete solution remained open.

keywords

  • Regular matrix pencils; Weierstrass canonical form; Low-rank perturbations; Matrix spectral perturbation theory; Partial multiplicities; Geometric approach