Two parameter eigenvalue problems for matrices (Q1115934)

The authors study the two-parameter eigenvalue problem \((T_ m- \lambda_ 1V_{m1}-\lambda_ 2V_{m2})x_ m=0\neq x_ m\), \(m=1,2\), where Tm, \(V_{mn}\) are Hermitian linear operators on the finite dimensional Hilbert space \(H_ m\). The usual definiteness assumption is replaced by a much weaker nonsingularity assumption, though it is assumed that there exists a definite linear combination of \(V_{m1}\) and \(V_{m2}\). Topics investigated include the change in algebraic and geometric eigenspaces and the transition between real and nonreal eigenvalues produced when a normalized version of the equations is embedded parametrically by applying a real origin shift to \(T_ 2\).