In Linear Algebra and its Applications by Lay, in chapter 5.5 on Complex Eigenvalues, regarding the rotation matrix you are given this theorem ( Theorem 9 ) :
Let A be a real 2 x 2 matrix with a complex eigenvalue $\lambda$ = a – bi ( b $\neq 0$) and an associated eigenvector v in $C^{2}$. Then
$A = PCP^{-1}$, where $P = [ Re\space v \space\space Im\space v]$ and $ C =
\begin{bmatrix}
a & -b \\
b & a \\
\end{bmatrix}
$
So the question is why is the eigenvalue with a – bi used and not a + bi, given that an array has two eigenvalues, where one is the complex conjugate of the other?
The book does not seem to mention this, if it does please correct me.
Best Answer
It's rather arbitrary. You could equally well use $\lambda=a+bi$ with associated eigenvector $v'$ and put the first column of $P$ as the real part of $v'$ and the second column as the negative of the imaginary part of $v'$, which would give exactly the same $P$ and $C.$ Indeed, it's even more arbirary. Your $b$ is not necessarily mine. If you say the eigenvalues are $11 \pm 17i$, so your $b$ is 17, I could equally well say the eigenvalues are $11 \pm (-17)i$ so my $b$ is -17. Then my $C$ would be different.Just writing an eigenvalue as $a+bi$ does not force $b$ to be positive.