Does quantum mechanics only deal with Hermitian/self-adjoint operators? Whatever matrix I am seeing in quantum mechanics are all Hermitian matrices. We are using their eigenvalues for different types of work. Fortunately all their eigenvalues are real. Have you ever seen any use of eigenvalues of real non-symmetric or non-Hermitian matrix in quantum mechanics? What are they? Is there any such? Please include some references.
 A: Unitary matrices are (almost) never Hermitian, hence no finite version of the infinitesimal transformations described by the Hermitian observables will ever be Hermitian itself.
The most prominent example is the time evolution $\mathcal{U}(t) = \mathrm{e}^{\mathrm{i}Ht}$, which is unitary, and not Hermitian.
A: If you want to describe the way in which a mixed state transforms, this in generally expressed using non-hermitian operators, either the Lindbladian in the continous case, or e.g. the Kraus representation of trace preserving completely positive maps in the discrete case.
A: Algebraic Quantum Field Theory starts from the assumption that a quantum mechanical system can be described by a C*-algebra. But not every element in the C*-algebra has a physical interpretation, but just the self-adjoint part, which has the structure of a Jordan algebra through the product
$$a\circ b = \frac{(a+b)^2 - a^2 - b^2}2.$$
The matrices you see arise from a representation of said C*-algebra, and of course representations of self-adjoint elements are self-adjoint (or "hermitian") matrices. Any other matrix has no "definite" physical meaning as something that you can measure.
