Some Results On Generalized Antieigenvalue Pairs and Their Associated Antieigenvectors of Certain Class of Linear Two-Parameter Eigenvalue Problems

In the field of operator algebra, the concept of antieigenvalue theory was first introduced by Karl Gustafson with a special emphasis on accretive operators. In a wide range of scientific applications, antieigenvalue naturally occurs. Due to the inclusion of nonlinear Euler equations in the antieigenvalue theory, computing antieigenvalues is a difficult task as compared to that of computing eigenvalues of the operator. In the current paper, we consider linear two-parameter eigenvalue problems (LTEP) and will discuss the abstract algebraic setting of the problem as proposed by Atkinson. We analyze the generalized antieigenvalue pair and their corresponding generalized antieigenvectors for LTEP using the consequences of the Cauchy Schwarz inequality. Some generalized antieigenvalue bounds will also be derived. Generalized antieigenvalues and their corresponding generalized antieigenvectors will be calculated solving their relevant optimization problem. For numerical computations, three examples will be provided. Real symmetric matrices are used in the first case, while real diagonal matrices are used in the second case and finally arbitrary matrices are considered.