Why are the eigenvalues of the symmetric matrix $A^TB^TBA$ equal to the eigenvalues of the matrix $B^TBAA^T$ where $A \in \mathbb{R}^{3 \times 3}$ and $B$ is either:
$B = \begin{pmatrix} {1}, {0}, {0} \end{pmatrix}$, $\begin{pmatrix} {1}, {0}, {0} \\ {0}, {0}, {1}\end{pmatrix}$ or $I_3$?
In fact, the nonzero eigenvalues of $CD$ and $DC$ are always the same for any matrices $C$ and $D$ that can be multiplied in both directions. See e.g. here. This is just the case $C = A^T$, $D = B^T B A$.