An operator acting on an eigenvector is just a constant times the vector. The constant is called the eigenvalue. The eigenvalues of hermitian operators are real. If two eigenvectors of a hermitian operator have different eigenvalues, then the vectors are orthogonal. For any finite dimensional vector space, it is always possible to form a complete orthonormal basis with eigenvectors of a hermitian operator. For an infinite dimensional vector space, it is not always possible. Those operators for which it is possible to form a complete orthonormal basis of eigenvectors are called observables - at least in this book. If two Hermitian operators commute, then it is possible to find a basis composed of vectors which are eigenvectors of both operators. If the operators do not commute, it is not possible to do this.