Self-adjoint
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In mathematics, an element x of a star-algebra is self-adjoint if <math>x^*=x<math>.
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if <math>x^*=y<math> then since <math>y^*=x^{**}=x<math> in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements.
In functional analysis, a linear operator A on a Hilbert space is called self-adjoint if it is equal to its own adjoint A*. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it agrees with its own conjugate transpose. Hermitian matrices are also called self-adjoint.
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