So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ and vectors $\bf v_i$. Therefore they will be symmetric and have an ON-basis of eigenvectors with real non-negative eigenvalues ( spectral theorem, right? ).
Now to my question. What would be a fast way to calculate a rotation matrix $\bf R$ such that $\| {\bf T}_1 - {\bf R} {\bf T}_2 {\bf R}^T \|_F^2$ is minimized?
Bonus points if one could easily incorporate some functionality which punishes large angles. Maybe a regularization like $\lambda \|{\bf R-I}\|_F^2$ or some more suitable one.
My own work so far is limited to realizing we can do eigenvalue decomposition of ${\bf T}_1$ and ${\bf T}_2$ and then sort eigenvalues and find rotation which pairwise maps the eigenvectors. But clearly that can not be the fastest way..?
The most you can hope to improve on the solution you found is a factor $2$. If you had a method better than that, you could use it to find the eigenvectors of $\mathbf T_2$ by finding the optimal $\mathbf R$ to align it with a non-degenerate diagonal matrix $\mathbf T_1$.