I found this observation in a book. "It is not possible to have an invariant definition of symmetry in one contravariant and one covariant index". That's all right, my problem is that to show how restrictive is to require symmetry on a mixed tensor $A^i_j$ the authors ask to resolve the following problem: If a tensor $A$ of type $(1,1)$ is symmetric in its indices with respect to every basis, that is, $A^i_j=A^j_i$, then $A$ is a multiple of the identity tensor, $A^i_j=\alpha\delta^i_j$.
I can prove that if $i\neq j\implies A^i_j=0$, but when $i=j$ I obtain different values instead of a constant value, in other words, the matrix $A^i_j$ is diagonal having different values along the diagonal.
Can anybody comment on this?