Normal matrices with orthogonal basis

Question

we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$. now my question is, how do i calculate this basis and the entries of the 2x2 matrices?

Answer 1

You could calculate an orthonormal basis of eigenvectors as a complex matrix. You can leave the real eigenvectors corresponding to real eigenvalues as they are. The non-real eiegnvalues occur in complex conjugate pairs. If $\lambda$ and $\overline{\lambda}$ are one such pair, and $\lambda = re^{i\theta},$ then you can replace the diagonal submatrix $\left( \begin{array}{clcr} \lambda & 0\\0 & \overline{\lambda} \end{array}\right)$ by the submatrix $\left( \begin{array}{clcr} r \cos \theta & r \sin \theta \\ -r \sin \theta & r \cos \theta\end{array}\right).$ Let $A$ be the real matrix you started with. Let $u,v$ be non-zero real vectors such that $A(u + iv) = \lambda (u + iv).$ Then comparing real and imaginary components shows that $Au = r \cos \theta u - r\sin \theta v , etc$. So, to obtain the corresponding real basis, you replace the pair of complex eigenvectors $u + iv, u-iv$ by the pair of real vectors $u,v$ in each case.