I suppose I have a real nonsymmetric diagonalizable matrix $A$ (for example $$\left(\begin{array}[cc]& 1 & 2\\0&-1\end{array}\right)$$).Then the eigenvectors split into right and left ones which are not transpose of the other: for the eigenvalue $\lambda$, $$x_L^T A=\lambda x_L^T$$ and $$A x_R=\lambda x_R$$
So the problem I have is that I don't know what the probability of measuring $\lambda$ when the system is in state $\Psi$ is, because : $|x_L^T\Psi|^2$ seems different from $|\Psi^T x_R|^2$.
The non symmetric matrix could for example arise in the following case : consider a symmetry of the system state : $R\Psi=\Psi$ and a symmetric operator $A'$, then $\langle A'\rangle=\Psi^T A'\Psi$. To obtain an asymmetry we "insert" $R$, instead of substituting, on the second $\Psi$, hence we get : $\Psi^T A' R\Psi$. $A'R$ could then be nonsymmetric if the symmetry does not commute with the measurement operator.
