In QM hermitean operators are used as observables, which provide real eigenvalues. According to a version of the spectral theorem, their eigenfunctions form an orthonormal basis. My question is about complex matrices and whether they might (may be under some restrictions) span also the entire space (i do not know exactly the spectral theorem). I try to understand the reasons for the usage of hermitean operators: in some references I found that they are used due to their real eigenvalues which correspond to measurements, but in others that they are used due to the fact that the eigenfunctions form an orthonormal basis. May be both?