2.18 ( ) Consider a real, symmetric matrix whose eigenvalue equation is given by (2.45). By taking the complex conjugate of this equation and

2.18 ( ) Consider a real, symmetric matrix Σ whose eigenvalue equation is given by (2.45). By taking the complex conjugate of this equation and subtracting the original equation, and then forming the inner product with eigenvector ui, show that the eigenvalues λi are real. Similarly, use the symmetry property of Σ to show that two eigenvectors ui and uj will be orthogonal provided λj = λi. Finally, show that without loss of generality, the set of eigenvectors can be chosen to be orthonormal, so that they satisfy (2.46), even if some of the eigenvalues are zero.