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.


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 $R_c$.


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