I'm reading a paper and came across the following line.
$R_g = R_{cg} R_c R_{cg}^{T}$
I recognize that $R_g$ and $R_c$ are similar matrices, and since they represent rotations, they are orthogonal.
As a result, the eigenvector matrix of $R_g$ can be transformed from that of $R_{c}$ using $R_{cg}$
I don't really understand the significance of the above statement nor how it came about. I understand that similar matrices share the same eigenvalues, but I don't know about any statements regarding eigenvectors.
Best Answer
Let $v$ be an eigenvector of $R_c$ with eigenvalue $\lambda$. Then since $R_{cg}$ are rotation matrices, $R^{-1}_{cg}=R^{T}_{cg}$, so that if $u:=R_{cg}v$, then $R_gu=R_{cg}R_cv=\lambda R_{cg}v = \lambda u$, so that $u$ is an eigenvector of $R_{g}$. All the above transformations are invertible, so that any eigenvector $u$ of $R_{g}$ is of the form $R_{cg}v$ for $v$ an eigenvector of $R_c$.