Hermitian operators are a special class of linear operators in quantum mechanics that have important properties. They are defined by the condition that the operator is equal to its own adjoint, meaning that the inner product of two vectors remains unchanged when the operator is applied. This characteristic ensures that the eigenvalues of Hermitian operators are real numbers, which is crucial for physical observables like position and momentum. In addition to their real eigenvalues, Hermitian operators have orthogonal eigenvectors, allowing for a complete basis in the corresponding vector space. This property is essential for the mathematical framework of quantum mechanics, as it enables the representation of quantum states and measurements. Examples of Hermitian operators include the position operator and the momentum operator.