A minimization problem

doodle

I have this matrix problem:

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

[tex]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[/tex]

is minimized, subject to the constraint [itex]V^T V = I[/itex].

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

[tex]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[/tex]

where [itex]\Lambda[/itex] 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

[tex] \Sigma_1 R_1 + \Sigma_2 R_2 + \Sigma_3 R_3[/tex]