# The nearest matrix over unit ball of matrix 2-norm

Given a matrix $$X$$, let $$\mathbf{proj}(X)=\underset{\|Y\|_2\le 1}{\arg\min} \|X-Y\|_F$$. Now the question is to solve $$\mathbf{proj}(X)$$.

Proposition: Suppose $$X=U \,\mathbf{diag}(\sigma)\,V^T$$ is the SVD of $$X$$, then $$\mathbf{proj}(X)=U\, \mathbf{diag}\bigl( \min(\sigma_1,1),\cdots,\min(\sigma_n,1) \bigr)\,V^T$$.

How to prove this proposition? My difficulty focus on proving that $$X$$ and $$\mathbf{proj}(X)$$ have the same singular vector.

• $\mathbf{proj}(X) = aX$ for some scalar $a$. – Paul Sinclair Apr 21 at 20:26