Linearizations of Cubic Two-Parameter Eigenvalue Problems and Its Vector Space

Abstract

The paper considers the study of linearization techniques of cubic two-parameter matrix polynomial (CTMP). We analyze vector spaces of linearization of CTMP, namely ansatz vector space and double ansatz vector space. We also consider cubic two-parameter eigenvalue problem (CTEP) to study their linearization classes. A unified framework on linearization of CTMP will be established. The conditions under which a matrix pencil in the ansatz spaces is a linearization of CTMP will also be derived. Moreover, using these linearization techniques, CTEP is first converted into a singular linear two-parameter eigenvalue problem (2PEP) of larger size so that existing numerical method for (2PEP) can be applied.