most recent 30 from http://mathoverflow.net 2013-05-19T04:32:08Z http://mathoverflow.net/feeds/user/12735 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54342/prove-if-a1-an-are-uniformly-distributed-unit-vectors-then-a1a1-anan/54344#54344 Alan 2011-02-04T18:33:41Z 2011-02-04T18:33:41Z

<p>This is false. </p> <p>If $a_1 + \cdots + a_n=0$ and if $a_1 a_1^T + \cdots a_n a_n^T = (1/2)I$, then $$(a_1 + \cdots + a_n)+(a_1+\cdots a_n)=0$$ and $$(a_1 a_1^T + \cdots a_n a_n^T)+(a_1 a_1^T + \cdots a_n a_n^T) = I.$$ So, having a sum of unit vectors be zero, does NOT imply that the sum of the matrices $a_i a_i^T$ is $(1/2)I$. </p> <p>You are missing a hypothesis.</p>