Find a complex matrix on a unit sub-spheres

I am new to optimization theory. I have a following question. For a given $$X = [x_1 x_2 \ldots x_N] \in \mathbb{C}^{N \times N}$$, where $$x_i \in \mathbb{C}^{N\times 1}$$ for $$i \in \{1,\ldots,N\}$$, $$U = [u_1 u_2 \ldots u_N]\in \mathbb{C}^{N \times N}$$, is there any possibility of obtaining a closed form solution of

$$\min_{U \in \mathbb{C}^{N\times N}}\{\|U-X\|_F^2: \|u_i\|_2^2=1\}$$

where $$\|\cdot\|_2$$ and $$\|\cdot\|_F$$ denotes the Euclidean and Frobenius norm respectively. I know how to find the solution for $$\|U\|_2^2=1$$, however, I do not see how to solve it when we have $$\|u_i\|^2=1$$.

I may be overthinking. I can just take the $$X$$ and normalize each column. I am not sure that would be the right approach.

• We have $\|U-X\|_F^2=\sum_i \|u_i-x_i\|_2^2$ and your approach would be correct. Apr 12, 2020 at 5:02
• Great! Thanks so much! Apr 12, 2020 at 5:42