Chapter 5.5, Problem 26E

Let A be a real 2 × 2 matrix with a complex eigenvalue λ = a − bi (b ≠ 0) and an associated eigenvector v in ℂ².

1. a. Show that A(Re v) = a Re v + b Im v and A(Im v) = −b Re v + a Im v. [Hint: Write v = Re v + i Im v, and compute Av.]
2. b. Verify that if P and C are given as in Theorem 9, then AP = PC.