I'm computing the eigenvectors of a real non-symmetric Matrix. I know that complex eigenvectors would come in conjugate. However, does anybody know a algorithm to rotate such complex eigenvectors that
- i) its real and imaginary parts are orthogonal
- ii) real part has norm 1 while imaginary parts has norm greater than 1.
The context is calculation of Empirical Normal Mode.
Say, I have a system evolve as $\frac{\mathrm{d}X}{\mathrm{d}t}=LX+\text{gaussian noise}$, $X$ is vector and $L$ is matrix.
Then the eigenvector of $L$ gives the "Empirical Normal Mode" and exponential of eigenvalues are corresponding timescale of each mode. In general, $X$ could be represented as linear combinations of Normal Modes. But $X$ is real then complex eigenvectors should appear in conjugate pair. I want to combine each complex conjugate pair in a unique way, that is, satisfy above two conditions.
Thanks!