I have learned in a previous question that a lower bound of $||T^{-1}AT||_F$ is $\sqrt{\sum_{i=1}^n|\lambda_i(A)|^2}$ for some $T$ and $\lambda_i(A)$ are the eigenvalues of $A$
Proof: Suppose $J=T_1^{-1}AT_1$ is the Jordan form of $A$. Let $T_2$=diag($1,k,k^2,...k^{n-1}$) where $k>0$. As $k\rightarrow \infty$ , $T_2^{-1}JT_2 \rightarrow $ diag($\lambda_1,...\lambda_n$) and thus the Frobenius norm approaches $\sqrt{\sum_{i=1}^n|\lambda_i(A)|^2}$ from above.
Thus I use $\sqrt{\sum_{i=1}^n|\lambda_i(A)|^2}$ as the result for
Given $A$
$\min_T ||T^{-1}AT||_F$
How could I go about finding a result for
Given $A$, $B$
$\min_T ||T^{-1}AT||_F + ||T^{-1}BT||_F$
Thanks in advance for any insight!
Link to previous question: Minimizing matrix norm via similarity transformation