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 $n>2$? 



Sums of Unitary Matrices
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](https://mathoverflow.net/questions/50936) 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.


