Optimizing Rank p Matrix V for Symmetric Matrices with SVD Using Frobenius Norm

[doodle]

I have this matrix problem:

Given $R_1, R_2, R_3\in\mathbb{R}^{N\times N}$ are symmetric matrices with rank $p<N$. Their SVD are $U_1\Sigma_1 U_1^T$, $U_2\Sigma_2 U_2^T$ and $U_3\Sigma_3 U_3^T$, respectively. I want to find a rank $p$ matrix $V$ such that

$$J = \|V\Sigma_1 V^T - U_1\Sigma_1 U_1^T\|_F^2 + \|V\Sigma_2 V^T - U_2\Sigma_2 U_2^T\|_F^2 + \|V\Sigma_3 V^T - U_3\Sigma_3 U_3^T\|_F^2$$

is minimized, subject to the constraint $V^T V = I$.

I tried using the trace for the Frobenius norm and ended up with

$$2V (\Sigma_1^2 + \Sigma_2^2 + \Sigma_3^2) - 4(U_1\Sigma_1 U_1^T V \Sigma_1 + U_2\Sigma_2 U_2^T V \Sigma_2 + U_3\Sigma_3 U_3^T V \Sigma_3) + V(\Lambda + \Lambda^T) = 0$$

where $\Lambda$ contains the Lagrange multipliers. I have no idea how to continue from here. Any help would be appreciated.

 

[doodle]

I take it that there is no simple solution here?

In the case where p = 1, the solution for V (when I tried to work it out) is the eigenvector corresponding to the largest eigenvalue of

$$\Sigma_1 R_1 + \Sigma_2 R_2 + \Sigma_3 R_3$$