=+19. In counting jumps in a Markov chain, it is possible to explicitly calculate the matrix exponential (13.12) for a two-state chain. Show that the eigenvalues of the matrix diag(Λ) + C(u) are

ω± = −(λ12 + λ21) ± (λ12 − λ21)2 + 4u2λ12λ21 2

when the donor and recipient subsets A and B equal {1, 2}. Argue that both eigenvalues are real and negative when u ∈ [0, 1). One is 0 and the other is negative when u = 1. Find the corresponding eigenvectors of the matrix diag(Λ) + C(u) for all u ∈ [0, 1]. What happens if A = {1} and B = {2} or vice versa?