In mathematics, a linear operator is called self-adjoint if it is equal to its own adjoint. This means that for any two vectors, the inner product of the operator applied to one vector and the other vector is the same as the inner product of the first vector and the operator applied to the second. Self-adjoint operators are important in various fields, including quantum mechanics and functional analysis. Self-adjoint operators have real eigenvalues and their eigenvectors corresponding to different eigenvalues are orthogonal. This property makes them particularly useful in solving differential equations and in the study of Hilbert spaces. In quantum mechanics, self-adjoint operators represent observable quantities, ensuring that measurement results are real numbers.