# minimize with orthogonal constraint

Let $\mathcal{L}=(L^2(\mathbb{R}^3))^N$ be the product space with the associated norm $$\Vert U\Vert_0=\left(\sum^N_{i=1}\Vert u_i\Vert_0^2\right)^{1/2}$$ where $U=(u_1,u_2,...,u_N)\in\mathcal{L}$. Let $$\mathcal{K}=\{U\in\mathcal{L};\int_{\mathbb{R}^3}u_iu_j=\delta_{i,j},\forall 1\leq i,j\leq N\}$$ and for any matrix $A\in\mathbb{R}^{N\times N}$ denote $$AU=(...,\sum_{j=1}^N A_{ij}U_j,...,)^T$$ For any $U_1,U_2\in\mathcal{K}$, is the following minimization problem have a unique solution $$\min\limits_{O\in\mathcal{O}^{N\times N}}\Vert OU_1-U_2\Vert_0$$ where $\mathcal{O}^{N\times N}$ is the set of all unitary matrix.

Thank you very much.

• Certainly not, as posed. Imagine the case when the whole set $U_1\cup U_2$ is orthonormal. Then the norm to minimize does not depend on $O$ at all. – fedja Jun 25 '17 at 19:46
• @fedja oh. I see. Thank you very much. I did not think about that. – whereamI Jun 25 '17 at 23:11