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The final property given above shows that if one views ''A'' as a [[linear operator]] from the Euclidean [[Hilbert space]] '''C'''<sup>''n''</sup> to '''C'''<sup>''m''</sup>, then the matrix ''A''<sup>*</sup> corresponds to the '''adjoint operator'''.
InFor factan itoperator can''A'' be used to define what is meant by that. Assuming now we are inon a [[Hilbert space]] ''H'', the relation
: <math> \langle A x, y \rangle = \langle x, A^* y \rangle </math>
<''Ax'',''y''> = <''x'', ''A''<sup>*</sup>''y''>
can be used to define the ''adjoint operator'' ''A''<sup>*</sup>, by means of the [[Riesz representation theorem]]. This definition can be extended even for operators which are not bounded. See [[self-adjoint operator]] for the details.
The notation <math>A^{\dagger}</math> is also used to denote the adjoint of ''A'', especially when used in conjunction with the [[bra-ket notation]]. The adjoint condition takes the form:
When working in [[Hilbert space]], especially with the [[bra-ket notation]], the '''adjoint operator''' - called the '''Hermitian Conjugate''', denoted as <math>A^{\dagger}</math>, is defined by the relation
:<math> \lang \phi | (A|\psi) \rang = \lang (A^{\dagger}\phi) | \psi) \rang </math>
The term '''Hermitian conjugate transpose''' is also sometimes used sinceto ifrefer <math>to Athe =adjoint. A^{\dagger} </math>Although thanthe Aetymology of this usage is callednot anclear, it has been suggested that it results from the expression '''[[Hermitian]] operator being used to denote self-adjoint operators, that is operators ''A'.' for which
:<math> A = A^{\dagger} </math>.
ItNote hasthat alsothere beenis stateda ingeneral which way the above can be related to the notion of a pairtheory of [[adjoint functor]]s in [[category theory]]. Anwhich explanationincludes isthe givenprevious bydefinition as a special case. See [[John Baez]]' inexpository article [http://math.ucr.edu/home/baez/week78.html week78] offor hisa famousdiscussion series.of Seethis, alsoand theearlier previous weekswritings for a gentleintroductory introductionmaterial toon category theory.
[[de:Adjungierte Matrix]]
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