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In [[mathematics]], an element ''x'' of a [[star-algebra]] is '''self-adjoint''' if  <math>x^*=x</math>.

A collection ''C'' of elements of a star-algebra is '''self-adjoint''' if it is closed under the [[Involution (mathematics)|involution]] operation. For example, if <math>x^*=y</math> then since <math>y^*=x^{**}=x</math> 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 operator|adjoint]] ''A''* and that the domain of ''A'' is the same as that of ''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 (mathematics)|matrix]] describing ''A'' with respect to this basis is [[Hermitian matrix|Hermitian]], i.e. if it is equal to its own [[conjugate transpose]]. Hermitian matrices are also called '''self-adjoint'''.

==See also==
*[[Symmetric matrix]]
*[[Hermitian]]

==References==
*{{cite book |authorlink=Michael C. Reed |first=M. |last=Reed |authorlink2=Barry Simon |first2=B. |last2=Simon |title=Methods of Mathematical Physics |others=Vol 2 |publisher=Academic Press |year=1972 |isbn= }}
*{{cite book |authorlink=Gerald Teschl |first=G. |last=Teschl |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |publisher=American Mathematical Society |location=Providence |year=2009 |url=http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ }}


{{DEFAULTSORT:Self-Adjoint}}
[[Category:Abstract algebra]]
[[Category:Linear algebra]]