Problem 6: Real Eigenvalue Decomposition Consider a real diagonalizable matrix A E Rx with, in general, complex eigenvectors and complex eigenvalues. Prove that if A e C is an eigenvalue of A associated with the eigenvector x e C", then A is an eigenvalue associated with the eigenvector i. What does this mean about the complex eigenvalues and eigenvectors of A? Prove that ifx = u + iv with u, v e R" and A = a + ib with a, b e R then A u v] = [u v] -b _ Problem 6: Real Eigenvalue Decomposition Consider a real diagonalizable matrix A E Rx with, in general, complex eigenvectors and complex eigenvalues. Prove that if A e C is an eigenvalue of A associated with the eigenvector x e C", then A is an eigenvalue associated with the eigenvector i. What does this mean about the complex eigenvalues and eigenvectors of A? Prove that ifx = u + iv with u, v e R" and A = a + ib with a, b e R then A u v] = [u v] -b _