A is real n×n,n≥2. I:every complex eigenvalue’s conjugate is also an eigenvalue. II:for a nonreal eigenvalue λ=x+iy,y≠0 and eigenvector v, Re v and Im v are independent over ℝ. Which hold?
Conjugating Av=λv gives A(conjugate v)=(conjugate λ)(conjugate v), proving I. If Re v and Im v were dependent, v would be a nonzero complex multiple of a nonzero real vector w, giving Aw=λw. The left side is real and λ has nonzero imaginary part, impossible for w≠0. Both true, **C**.
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