At least one diagonal element of any real symmetric matrix of rank $1$ is non-zero ? If $A$ is a real symmetric matrix of rank $1$ then is it true that at least one diagonal element is  non-zero ?

a rank one symmetric matrix is of the form $aa^T$ for some nonzero vector $a.$ the $i$th diagonal element of $aa^T$ is $a_i^2$. so $a \neq 0$ implies at least one of the diagonal elements is nonzero. 
