(a) Prove that if x ± iy is a complex conjugate pair of eigenvectors of a real matrix A corresponding to complex conjugate eigenvalues μ ± iv with v ≠ 0, then x and y are linearly independent real vectors.
(b) More generally, if vj = xj ± iyj, j = 1, ..., k are complex conjugate pairs of eigenvectors corresponding to distinct pairs of complex conjugate eigenvalues μj ± i vj, vj ≠ 0, then the real vectors x1, ..., xk, y1, ..., yk are linearly independent.