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# Science Fair Project Encyclopedia

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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 {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.