Sums of Unitary Matrices Let $J$ be the $n$ by $n$ matrix whose each entry is $1$. Also define $f(n)$ to be the least $m$ so that there is a $\lambda>0$ so that $\lambda J$ is the sum of at most $m$ unitary matrices. Note $f(2)=2$. 
What is the value of $f(n)$ for $n>2$? 
 A: Edit: I correct the mistake pointed out by Matthew in his comment.
In fact, any matrix $X$ can be written as $X=\lambda(U+V)$ for unitary matrices $U,V$, and $\lambda$ can be taken as a half of the operator norm of $X$. For a proof, see this question. This is slightly stronger than Matthew's comment, and the proof works in any finite von Neumann algebra (i.e. when the partial isometry in the polar decomposition can be taken as a unitary). In a $C^*$-algebra this is not possible (consider $z \mapsto z$ in the $C^*$-algebra of continuous functions on the unit disc of the complex plane) but, as noted by Matthew, you can get the same decomposition with $4$ unitaries.
A: Just for fun ... This is more a comment than an answer. If $n=2$, the unit sphere $\Sigma$ of $M_2(\mathbb R)$ (the set of matrices whose operator norm equals $1$) is the joint of $SO_2$ and of $O_2^-$ (the set of orthogonal symmetries). The joint of two subsets $X$ and $Y$ is by definition the union of the segments $[x,y]$ with $x\in X$ and $y\in Y$. In particular, if $A\in\Sigma\setminus O_2$, then it is a unique convex combination of a rotation and a symmetry.
More interestingly, both $SO_2$ and $O_2^-$ are circles in the $3$-dimensional sphere $\Sigma$, and they are linked. In other word, every disk $D\in\Sigma$ with boundary $SO_2$ respectively $O_2^-$) must intersect $O_2^-$ (resp. $SO_2$).
