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?
['https://physics.stackexchange.com/questions/718982', 'https://physics.stackexchange.com', 'https://physics.stackexchange.com/users/80627/']
Consider the change of basis with the matrix $\begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix}$.
$$
\begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0\\
0 & a
\end{pmatrix}
\begin{pmatrix}
1 & -1\\
0 & 1
\end{pmatrix}
=
\begin{pmatrix}
1 & a-1\\
0 & a
\end{pmatrix}
$$
is symmetric if and only if $a=1$. In more than two dimensions just use this change of basis in a two-dimensional subspace and the identity in the normal space to it.
Thank you, guys. I think you both tackled special cases. However, your matrix approaches led me to a general solution I believe is also correct.
From the general transformation $A'=S^{-1}AS$ we have $SA'=AS$. Assuming we already proved that $A$ is diagonal, let $A=diag\{\lambda\_1\ldots\lambda\_n\}$ and $A'=diag\{\lambda'\_1\ldots\lambda'\_n\}$, then we have
$$(SA')\_{ik}=\sum\_j s\_{ij}a'\_{jk}=s\_{ik}\lambda'\_k$$
$$(AS)\_{ik}=\sum\_j a\_{ij}s\_{jk}=\lambda\_i s\_{ik}$$
Since the problem states the condition must hold for arbitrary bases, we can assume that for a given $k,\quad s\_{ik}\neq0$ for al $i$, then we have
$$\lambda\_i=\lambda'\_k=\alpha,\qquad i=1,2,\ldots,n$$
Since $A^i\_j=0$ for $i\neq j$ we have $A^i\_j=\alpha\delta^i\_j$