Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $J$ be the $n$ by $n$ matrix of all 1's. Let $f(n)$ be the least number $m$ of unitary matrices $U_1,\dots,U_m$ so that $J = U_1 + \cdots + U_m$. What can you say about the growth of the function $f(n)$?

share|cite|improve this question

Hint: $J$ has an eigenvector for eigenvalue $n$. What can you say about $\|Uv\|$ for a unitary matrix $U$ and vector $v$?

share|cite|improve this answer

Robert Israel has given a hint for finding a lower bound on $f(n)$. To see that the lower bound is sharp, you can write $J$ as a sum of permutation matrices.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.