Self-adjoint


Self-adjoint

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 is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint.

ee also

*symmetric matrix
*Hermitian


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