On the Kronecker Structure of linearization of Cubic Two-Parameter Eigenvalue Problems

Linearization is a classical approach to study matrix polynomial of the form P(lambda)=Sum lambdaj Aj, where A j Cnxn . It converts into a matrix pencil of the form L(lambda)=A+lambda B of high dimension, where A and B are matrices over C , and lambda is the spectral parameter. In this paper we consider Cubic two-parameter eigenvalue problems ( CTEP) and study three different linearization process of the problem. Usinglinearization techniques, CTEP is first converted into a linear two-parameter eigenvalue problem(L2EP) of coefficient matrices of different sizes. The main advantage of these linearizations lies in the fact that, after transforming them into suitable linearized forms, existing numerical techniques for linear multiparameter eigenvalue problems (LMEP) can be applied to CTEP without solving the original problem. While solving CTEP by formulating suitable linearizations, several transformations are generally used. In the current paper, it is also intended to report on these transformations, which have not been studied completely due to the complexity of their Kronecker structures.