At least one diagonal element of any real symmetric matrix of rank $1$ is non-zero ?

Question

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

Answer 1

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 $\operatorname{tr} A = \operatorname{tr} \Delta$.

Answer 2

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.