selfadjoint

Self-adjointness is a property of linear operators in linear algebra and functional analysis. An operator A is said to be self-adjoint if it is equal to its adjoint, denoted as A = A*. The adjoint of an operator A, denoted A*, is defined such that for any vectors x and y in the domain of A, the following equation holds: <Ax, y> = <x, A*y>, where <, > denotes the inner product. Self-adjoint operators play a crucial role in various areas of mathematics and physics, particularly in quantum mechanics, where they are often used to describe observable quantities. One of the key properties of self-adjoint operators is that their spectrum consists entirely of real numbers. This means that all eigenvalues of a self-adjoint operator are real. Additionally, self-adjoint operators are closed and densely defined, which makes them well-behaved in many mathematical contexts. In finite-dimensional spaces, self-adjoint operators correspond to symmetric matrices, and their eigenvalues are real. In infinite-dimensional spaces, the concept of self-adjointness becomes more complex, but it remains a fundamental notion in the study of unbounded operators and spectral theory.