Question: Let \(A \in M_{n}(\mathbf{R})\). Suppose that \(\lambda\) is a real eigenvalue of \(A\) and that \(A x=\lambda x, x \in\) \(\mathbf{C}^{n}, x \neq 0 .\) Let \(x=u+i v\), in which \(u, v \in \mathbf{R}^{n}\) are the respective real and imaginary parts of $x ;$ see \((0.2 .5) .\) Show that \(A u=\lambda u\) and \(A v=\lambda v .\) Explain why at least one of $u, v$ must be nonzero, and conclude that \(A\) has a real eigenvector associated with \(\lambda\). Must both \(u\) and \(v\) be eigenvectors of A ? Can \(A\) have a real eigenvector associated with an eigenvalue that is not real?

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