TY - JOUR

T1 - On linearizations of the quadratic two-parameter eigenvalue problems

AU - Hochstenbach, M.E.

AU - Muhic, A.

AU - Plestenjak, Bor

PY - 2012

Y1 - 2012

N2 - We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparameter eigenvalue problem. Two of these transformations are generalizations of the well-known linearization of the quadratic eigenvalue problem and linearize the QMEP as a singular two-parameter eigenvalue problem. The third replaces all nonlinear terms by new variables and adds new equations for their relations. The QMEP is thus transformed into a nonsingular five-parameter eigenvalue problem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparameter eigenvalue problems. We also consider several special cases of the QMEP, where some matrix coefficients are zero.

AB - We present several transformations that can be used to solve the quadratic two-parameter eigenvalue problem (QMEP), by formulating an associated linear multiparameter eigenvalue problem. Two of these transformations are generalizations of the well-known linearization of the quadratic eigenvalue problem and linearize the QMEP as a singular two-parameter eigenvalue problem. The third replaces all nonlinear terms by new variables and adds new equations for their relations. The QMEP is thus transformed into a nonsingular five-parameter eigenvalue problem. The advantage of these transformations is that they enable one to solve the QMEP using existing numerical methods for multiparameter eigenvalue problems. We also consider several special cases of the QMEP, where some matrix coefficients are zero.

U2 - 10.1016/j.laa.2011.07.026

DO - 10.1016/j.laa.2011.07.026

M3 - Article

SN - 0024-3795

VL - 436

SP - 2725

EP - 2743

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 8

ER -