Singular two-parameter eigenvalue problems
Bor Plestenjak · 2010
A1x1 = λB1x1 + μC1x1, (1) A2x2 = λB2x2 + μC2x2, where Ai, Bi, and Ci are given ni × ni complex matrices, λ, μ ∈ C, and xi ∈ Ci for i = 1, 2. A pair (λ, μ) is an eigenvalue if it satisfies (1) for nonzero vectors x1, x2, and the corresponding eigenvector is x1 ⊗ x2. On the tensor product space we can define n1n2 × n1n2 matrices ∆0 = B1 ⊗ C2 − C1 ⊗B2, ∆1 = A1 ⊗ C2 − C1 ⊗ A2, ∆2 = B1 ⊗ A2 − A1 ⊗B2. The two-parameter eigenvalue problem (1) is nonsingular if its operator determinant ∆0 is invertible. Atkinson showed [1] that a nonsingular two-parameter eigenvalue problem is equivalent to the joint generalized eigenvalue problems