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 ?

The matrix $A$ can be diagonalized: the diagonalized matrix $\Delta$ has real eigenvalues, and only one is non-zero (it has rank 1 as well). Thus, $\operatorname{tr} \Delta \neq 0$. Yet, the trace is invariant, so $\operatorname{tr} A = \operatorname{tr} \Delta$.
