Unitary transformation

<h2 id="formal-definition">Formal definition</h2>
<p>More precisely, a unitary transformation is an <a href="/facts/Isometry/K5wX8ah6">isometric isomorphism</a> between two <a href="/facts/Inner_product_space/HoyjElEy">inner product spaces</a> (such as <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>). In other words, a <i>unitary transformation</i> is a <a href="/facts/Bijective_function/j4gTuTmW">bijective function</a>
</p>

U
        :
        
          H
          
            1
          
        
        →
        
          H
          
            2
          
        
      
    
    {\displaystyle U:H_{1}\to H_{2}}

<p>between two inner product spaces, 
  
    
      
        
          H
          
            1
          
        
      
    
    {\displaystyle H_{1}}
  
 and 
  
    
      
        
          H
          
            2
          
        
        ,
      
    
    {\displaystyle H_{2},}
  
 such that 
</p>

⟨
        U
        x
        ,
        U
        y
        
          ⟩
          
            
              H
              
                2
              
            
          
        
        =
        ⟨
        x
        ,
        y
        
          ⟩
          
            
              H
              
                1
              
            
          
        
        
        
           for all 
        
        x
        ,
        y
        ∈
        
          H
          
            1
          
        
        .
      
    
    {\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}

<p>It is a <a href="/facts/Linear_isometry/K5wX8ah6">linear isometry</a>, as one can see by setting 
  
    
      
        x
        =
        y
        .
      
    
    {\displaystyle x=y.}

</p>
<h2 id="unitary-operator">Unitary operator</h2>
<p>In the case when 
  
    
      
        
          H
          
            1
          
        
      
    
    {\displaystyle H_{1}}
  
 and 
  
    
      
        
          H
          
            2
          
        
      
    
    {\displaystyle H_{2}}
  
 are the same space, a unitary transformation is an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> of that Hilbert space, and then it is also called a <a href="/facts/Unitary_operator/rkypfQQ9">unitary operator</a>.
</p>
<h2 id="antiunitary-transformation">Antiunitary transformation</h2>
<p>A closely related notion is that of <a href="/facts/Antiunitary/sKtIU7nS">antiunitary</a> transformation, which is a bijective function
</p>

U
        :
        
          H
          
            1
          
        
        →
        
          H
          
            2
          
        
        
      
    
    {\displaystyle U:H_{1}\to H_{2}\,}

<p>between two <a href="/facts/Complex_number/2w7mqI7e">complex</a> Hilbert spaces such that 
</p>

⟨
        U
        x
        ,
        U
        y
        ⟩
        =
        
          
            
              ⟨
              x
              ,
              y
              ⟩
            
            ¯
          
        
        =
        ⟨
        y
        ,
        x
        ⟩
      
    
    {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }

<p>for all 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 in 
  
    
      
        
          H
          
            1
          
        
      
    
    {\displaystyle H_{1}}
  
, where the horizontal bar represents the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a>. 
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Antiunitary/sKtIU7nS">Antiunitary</a></li>
<li><a href="/facts/Orthogonal_transformation/s7LGBDFX">Orthogonal transformation</a></li>
<li><a href="/facts/T-symmetry/Tgf2LJMc">Time reversal</a></li>
<li><a href="/facts/Unitary_group/lEvudIhU">Unitary group</a></li>
<li><a href="/facts/Unitary_operator/rkypfQQ9">Unitary operator</a></li>
<li><a href="/facts/Unitary_matrix/8fsxv3KD">Unitary matrix</a></li>
<li><a href="/facts/Wigner%27s_theorem/IHmtm9bF">Wigner's theorem</a></li>
<li><a href="/facts/Unitary_transformation_(quantum_mechanics)/DCzYRCxK">Unitary transformations in quantum mechanics</a></li></ul>

Unitary transformation open-in-new

Unitary transformation