# Extensive Definition

In mathematics, an element x of
a star-algebra
is self-adjoint if x^*=x.

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 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.

## See also

selfadjoint in German:
Selbstadjungiert