Self-adjoint Form

In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if $x^*=x$. 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 $x^*=y$ then since $y^*=x^=x$ in a star-algebra, the set is a self-adjoint set even though ''x'' and ''y'' need not be self-adjoint elements.

In functional analysis, a linear operator $A : H \to H$ 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 $f$ is called self-adjoint if $f=f^\dagger$; this is possible only for an endomorphism $f\colon a \to a$.

See also

* Hermitian matrix
* Normal element
* Symmetric matrix
* Self-adjoint operator
* Unitary element

References

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{{DEFAULTSORT:Self-Adjoint
Abstract algebra
Linear algebra