Abstract algebra | Linear algebra

Self-adjoint

In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since 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 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 is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger category, a morphism is called self-adjoint if ; this is possible only for an endomorphism . (Wikipedia).

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Related pages

Dagger category | Hilbert space | Abstract algebra | Functional analysis | Normal element | Mathematics | Unitary element | Self-adjoint operator | Conjugate transpose | *-algebra | Orthonormal basis | Matrix (mathematics) | Hermitian matrix | Involution (mathematics) | Endomorphism | Symmetric matrix | Morphism