TY - JOUR
T1 - On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure
AU - Moro, Julio
AU - Burke, James V.
AU - Overton, Michael L.
PY - 1997/10
Y1 - 1997/10
N2 - Let A be a complex matrix with arbitrary Jordan structure and λ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S.R. Comput. Math. and Math. Phys., 1 (1965), pp. 73-85], showing that the splitting of λ under a small perturbation of A of order ε is, generically, of order ε1/n. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.
AB - Let A be a complex matrix with arbitrary Jordan structure and λ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S.R. Comput. Math. and Math. Phys., 1 (1965), pp. 73-85], showing that the splitting of λ under a small perturbation of A of order ε is, generically, of order ε1/n. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.
KW - Newton diagram
KW - Newton envelope
KW - Perturbation of eigenvalues
KW - Perturbation theory for linear operators
KW - Spectral condition number
KW - Stability theory
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U2 - 10.1137/S0895479895294666
DO - 10.1137/S0895479895294666
M3 - Article
AN - SCOPUS:0031497208
SN - 0895-4798
VL - 18
SP - 793
EP - 817
JO - SIAM Journal on Matrix Analysis and Applications
JF - SIAM Journal on Matrix Analysis and Applications
IS - 4
ER -