Let $\psi^{A \tilde{A}}:=|\psi\rangle\langle\psi|$ with $\left|\psi^{A \tilde{A}}\right\rangle=\sum_{x=1}^{|A|} \sqrt{p_{x}}|x\rangle|x\rangle \in A \otimes \tilde{A}$.
Now, I'm looking for the eigenvalues and eigenvectors of $\mathrm{id}^{A} \otimes \mathcal{T}\left(\psi^{A \tilde{A}}\right),$ where $\mathcal{T}$ is the transpose map.
ATTEMPT:
First I find out that the eigenvalues of $\psi^{A \tilde{A}}$ is one and $|A|$ zeros if $\sqrt{p_{x}}=\frac{1}{|A|}$. But I have no idea about this problem which $\sqrt{p_{x}}$ can be anything. Totally I know that a rank-$1$ matrix has one none-zero eigenvalue (if you ask me why, I'll tell you :-|).
But the main part of my question is about the eigenvalues and eigenvectors of $\mathrm{id}^{A} \otimes \mathcal{T}\left(\psi^{A \tilde{A}}\right),$. I think this is a tensor commutation matrix $|A|\otimes |A|$. Could you please help me?
I should mention that hilbert space $\tilde{A}$ is replica of $A$ and they are exactly the same. And $\mathrm{id}^{A}$ is the identity matrix in $A$