Given a set of real symmetric matrices $\boldsymbol{A}_1,\dots,\boldsymbol{A}_n$. I need to find the orthogonal matrix $\boldsymbol{\Phi}$ that solves:
$\min_{\boldsymbol{\Phi}}\sum_i \|$zero_diag($\boldsymbol{\Phi}^T \boldsymbol{A}_i \boldsymbol{\Phi}$) $\|_F^2$ s.t. $\boldsymbol{\Phi}^T \boldsymbol{\Phi} = \boldsymbol{\Phi} \boldsymbol{\Phi}^T = \boldsymbol{I}$.
Here, zero_diag does the following: 1. takes a matrix, 2. zeros its diagonal entries, 3. returns the resulted matrix. For example: zero_diag($[1 \, 2; 3 \, 4]$)=$[0 \, 2; 3 \, 0]$.
Is there a closed form for $\boldsymbol{\Phi}$ (perhaps in terms of eigenvectors of matrices)?