Hermitian adjoint

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>, each <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 on an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> defines a Hermitian adjoint (or adjoint) operator 
 
 
 
 
 A
 
 ∗
 
 
 
 
 {\displaystyle A^{*}}
 
 on that space according to the rule

⟨
        A
        x
        ,
        y
        ⟩
        =
        ⟨
        x
        ,
        
          A
          
            ∗
          
        
        y
        ⟩
        ,
      
    
    {\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}

where 
 
 
 
 ⟨
 ⋅
 ,
 ⋅
 ⟩
 
 
 {\displaystyle \langle \cdot ,\cdot \rangle }
 
 is the <a href="/facts/Inner_product/HoyjElEy">inner product</a> on the <a href="/facts/Vector_space/M1pxMLx2">vector space</a>.
The adjoint may also be called the Hermitian conjugate or simply the Hermitian after <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a>. It is often denoted by A† in fields like <a href="/facts/Physics/a7aH2y08">physics</a>, especially when used in conjunction with <a href="/facts/Bra%E2%80%93ket_notation/NDlKsaRh">bra–ket notation</a> in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. In <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite dimensions</a> where operators can be represented by <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a>, the Hermitian adjoint is given by the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> (also known as the Hermitian transpose).
The above definition of an adjoint operator extends verbatim to <a href="/facts/Bounded_operator/LYmDTIqR">bounded linear operators</a> on <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> 
 
 
 
 H
 
 
 {\displaystyle H}
 
. The definition has been further extended to include unbounded <a href="/facts/Densely_defined_operator/8uGARqJR">densely defined</a> operators, whose domain is topologically <a href="/facts/Dense_(topology)/nPT9Yxao">dense</a> in, but not necessarily equal to, 
 
 
 
 H
 .
 
 
 {\displaystyle H.}

Hermitian adjoint open-in-new

Hermitian adjoint